The Congruent Number Problem Elliptic Curves

Webinar Organized by Congruent Numbers IQAC & Dept. of Maths., Lakhimpur Girls’ College with Gonit Sora Elliptic Curves July 10, 2020

The Congruent Number Problem and Elliptic Curves

Anupam Saikia Department of Mathematics, Indian Institute of Technology Guwahati

Anupam Saikia IIT Guwahati July 11, 2020 1 / 28 Sections

Congruent Numbers

Elliptic Curves

1 Congruent Numbers

2 Elliptic Curves

Anupam Saikia IIT Guwahati July 11, 2020 1 / 28 Sections

Congruent Numbers

Elliptic Curves

1 Congruent Numbers

2 Elliptic Curves

Anupam Saikia IIT Guwahati July 11, 2020 2 / 28 Number Theory

Number Theory is the branch of mathematics that studies hidden

Congruent properties of integers and rational numbers. Numbers

Elliptic Curves A charm of Number Theory is that it has many statements that can be easily appreciated through examples but difficult to prove. E.g., (Goldbach Conjecture, 1742) Every even natural number n ≥ 4 can be written as the sum of two primes. E.g., 4=2+2 , 6=3+3 , 8=3+5 , ···. (Weak Goldbach Conjecture) Every odd number greater than 5 can be expressed as the sum of 3 primes. Note that (2n + 1) − 3 = p + q (Goldbach) implies 2n + 1 = 3 + p + q (Weak Goldbach). For example, 21 − 3 = 18 = 7 + 11 implies 21 = 3 + 7 + 11. Significant contribution by Hardy-Littlewood (1923), Vinogradov (1935), Ramare (1995), Terrence Tao (2012), Helfgott (2013) toward the Weak Goldbach Conjecture.

Anupam Saikia IIT Guwahati July 11, 2020 3 / 28 Number Theory

Number Theory is the branch of mathematics that studies hidden

Congruent properties of integers and rational numbers. Numbers

Elliptic Curves A charm of Number Theory is that it has many statements that can be easily appreciated through examples but difficult to prove. E.g., (Goldbach Conjecture, 1742) Every even natural number n ≥ 4 can be written as the sum of two primes. E.g., 4=2+2 , 6=3+3 , 8=3+5 , ···. (Weak Goldbach Conjecture) Every odd number greater than 5 can be expressed as the sum of 3 primes. Note that (2n + 1) − 3 = p + q (Goldbach) implies 2n + 1 = 3 + p + q (Weak Goldbach). For example, 21 − 3 = 18 = 7 + 11 implies 21 = 3 + 7 + 11. Significant contribution by Hardy-Littlewood (1923), Vinogradov (1935), Ramare (1995), Terrence Tao (2012), Helfgott (2013) toward the Weak Goldbach Conjecture.

Anupam Saikia IIT Guwahati July 11, 2020 3 / 28 Number Theory

Number Theory is the branch of mathematics that studies hidden

Congruent properties of integers and rational numbers. Numbers

Elliptic Curves A charm of Number Theory is that it has many statements that can be easily appreciated through examples but difficult to prove. E.g., (Goldbach Conjecture, 1742) Every even natural number n ≥ 4 can be written as the sum of two primes. E.g., 4=2+2 , 6=3+3 , 8=3+5 , ···. (Weak Goldbach Conjecture) Every odd number greater than 5 can be expressed as the sum of 3 primes. Note that (2n + 1) − 3 = p + q (Goldbach) implies 2n + 1 = 3 + p + q (Weak Goldbach). For example, 21 − 3 = 18 = 7 + 11 implies 21 = 3 + 7 + 11. Significant contribution by Hardy-Littlewood (1923), Vinogradov (1935), Ramare (1995), Terrence Tao (2012), Helfgott (2013) toward the Weak Goldbach Conjecture.

Anupam Saikia IIT Guwahati July 11, 2020 3 / 28 Number Theory

Number Theory is the branch of mathematics that studies hidden

Congruent properties of integers and rational numbers. Numbers

Elliptic Curves A charm of Number Theory is that it has many statements that can be easily appreciated through examples but difficult to prove. E.g., (Goldbach Conjecture, 1742) Every even natural number n ≥ 4 can be written as the sum of two primes. E.g., 4=2+2 , 6=3+3 , 8=3+5 , ···. (Weak Goldbach Conjecture) Every odd number greater than 5 can be expressed as the sum of 3 primes. Note that (2n + 1) − 3 = p + q (Goldbach) implies 2n + 1 = 3 + p + q (Weak Goldbach). For example, 21 − 3 = 18 = 7 + 11 implies 21 = 3 + 7 + 11. Significant contribution by Hardy-Littlewood (1923), Vinogradov (1935), Ramare (1995), Terrence Tao (2012), Helfgott (2013) toward the Weak Goldbach Conjecture.

Anupam Saikia IIT Guwahati July 11, 2020 3 / 28 Number Theory

Number Theory is the branch of mathematics that studies hidden

Congruent properties of integers and rational numbers. Numbers

Elliptic Curves A charm of Number Theory is that it has many statements that can be easily appreciated through examples but difficult to prove. E.g., (Goldbach Conjecture, 1742) Every even natural number n ≥ 4 can be written as the sum of two primes. E.g., 4=2+2 , 6=3+3 , 8=3+5 , ···. (Weak Goldbach Conjecture) Every odd number greater than 5 can be expressed as the sum of 3 primes. Note that (2n + 1) − 3 = p + q (Goldbach) implies 2n + 1 = 3 + p + q (Weak Goldbach). For example, 21 − 3 = 18 = 7 + 11 implies 21 = 3 + 7 + 11. Significant contribution by Hardy-Littlewood (1923), Vinogradov (1935), Ramare (1995), Terrence Tao (2012), Helfgott (2013) toward the Weak Goldbach Conjecture.

Anupam Saikia IIT Guwahati July 11, 2020 3 / 28 Number Theory

Number Theory is the branch of mathematics that studies hidden

Congruent properties of integers and rational numbers. Numbers

Elliptic Curves A charm of Number Theory is that it has many statements that can be easily appreciated through examples but difficult to prove. E.g., (Goldbach Conjecture, 1742) Every even natural number n ≥ 4 can be written as the sum of two primes. E.g., 4=2+2 , 6=3+3 , 8=3+5 , ···. (Weak Goldbach Conjecture) Every odd number greater than 5 can be expressed as the sum of 3 primes. Note that (2n + 1) − 3 = p + q (Goldbach) implies 2n + 1 = 3 + p + q (Weak Goldbach). For example, 21 − 3 = 18 = 7 + 11 implies 21 = 3 + 7 + 11. Significant contribution by Hardy-Littlewood (1923), Vinogradov (1935), Ramare (1995), Terrence Tao (2012), Helfgott (2013) toward the Weak Goldbach Conjecture.

Anupam Saikia IIT Guwahati July 11, 2020 3 / 28 Number Theory

Number Theory is the branch of mathematics that studies hidden

Congruent properties of integers and rational numbers. Numbers

Elliptic Curves A charm of Number Theory is that it has many statements that can be easily appreciated through examples but difficult to prove. E.g., (Goldbach Conjecture, 1742) Every even natural number n ≥ 4 can be written as the sum of two primes. E.g., 4=2+2 , 6=3+3 , 8=3+5 , ···. (Weak Goldbach Conjecture) Every odd number greater than 5 can be expressed as the sum of 3 primes. Note that (2n + 1) − 3 = p + q (Goldbach) implies 2n + 1 = 3 + p + q (Weak Goldbach). For example, 21 − 3 = 18 = 7 + 11 implies 21 = 3 + 7 + 11. Significant contribution by Hardy-Littlewood (1923), Vinogradov (1935), Ramare (1995), Terrence Tao (2012), Helfgott (2013) toward the Weak Goldbach Conjecture.

Anupam Saikia IIT Guwahati July 11, 2020 3 / 28 Number Theory

Number Theory is the branch of mathematics that studies hidden

Congruent properties of integers and rational numbers. Numbers

Elliptic Curves A charm of Number Theory is that it has many statements that can be easily appreciated through examples but difficult to prove. E.g., (Goldbach Conjecture, 1742) Every even natural number n ≥ 4 can be written as the sum of two primes. E.g., 4=2+2 , 6=3+3 , 8=3+5 , ···. (Weak Goldbach Conjecture) Every odd number greater than 5 can be expressed as the sum of 3 primes. Note that (2n + 1) − 3 = p + q (Goldbach) implies 2n + 1 = 3 + p + q (Weak Goldbach). For example, 21 − 3 = 18 = 7 + 11 implies 21 = 3 + 7 + 11. Significant contribution by Hardy-Littlewood (1923), Vinogradov (1935), Ramare (1995), Terrence Tao (2012), Helfgott (2013) toward the Weak Goldbach Conjecture.

Anupam Saikia IIT Guwahati July 11, 2020 3 / 28 The Congruent Number Problem

“The Congruent Number Problem, the written history of Congruent Numbers which can be traced back at least a millennium, is the oldest Elliptic Curves unresolved major problem in number theory, and perhaps in the whole of mathematics.” John Coates (PNAS, 2012)

The Congruent Number Problem can now be thought of as the oldest manifestation of a famous conjecture known as the Birch and Swinnerton-Dyer (BSD) Conjecture.

The BSD Conjecture was included in the seven millennium problems (2000), and is yet to be resolved completely.

A century before that, Hilbert posed 23 problems which greatly influenced mathematics in the 20-th century.

Anupam Saikia IIT Guwahati July 11, 2020 4 / 28 The Congruent Number Problem

“The Congruent Number Problem, the written history of Congruent Numbers which can be traced back at least a millennium, is the oldest Elliptic Curves unresolved major problem in number theory, and perhaps in the whole of mathematics.” John Coates (PNAS, 2012)

The Congruent Number Problem can now be thought of as the oldest manifestation of a famous conjecture known as the Birch and Swinnerton-Dyer (BSD) Conjecture.

The BSD Conjecture was included in the seven millennium problems (2000), and is yet to be resolved completely.

A century before that, Hilbert posed 23 problems which greatly influenced mathematics in the 20-th century.

Anupam Saikia IIT Guwahati July 11, 2020 4 / 28 The Congruent Number Problem

“The Congruent Number Problem, the written history of Congruent Numbers which can be traced back at least a millennium, is the oldest Elliptic Curves unresolved major problem in number theory, and perhaps in the whole of mathematics.” John Coates (PNAS, 2012)

The Congruent Number Problem can now be thought of as the oldest manifestation of a famous conjecture known as the Birch and Swinnerton-Dyer (BSD) Conjecture.

The BSD Conjecture was included in the seven millennium problems (2000), and is yet to be resolved completely.

A century before that, Hilbert posed 23 problems which greatly influenced mathematics in the 20-th century.

Anupam Saikia IIT Guwahati July 11, 2020 4 / 28 The Congruent Number Problem

“The Congruent Number Problem, the written history of Congruent Numbers which can be traced back at least a millennium, is the oldest Elliptic Curves unresolved major problem in number theory, and perhaps in the whole of mathematics.” John Coates (PNAS, 2012)

The Congruent Number Problem can now be thought of as the oldest manifestation of a famous conjecture known as the Birch and Swinnerton-Dyer (BSD) Conjecture.

The BSD Conjecture was included in the seven millennium problems (2000), and is yet to be resolved completely.

A century before that, Hilbert posed 23 problems which greatly influenced mathematics in the 20-th century.

Anupam Saikia IIT Guwahati July 11, 2020 4 / 28 What Is a Congruent Number

Let Z = {0, ±1, ±2,...}, the set of integers, and

Congruent m Numbers Q = { n | m, n are inetgers, n 6= 0}, the set of rational numbers. Elliptic Curves A positive integer n is called congruent if it is the area of a right triangle all of which sides are rational numbers, i.e., 1 n = pq, p2 + q2 = r2, p, q, r ∈ . 2 Q

r   q p, q, r ∈ Q (a P ythagorean triple) 1  n = Area = 2 ×p×q p

Example: 6 is a congruent number as 1 6 = .3.4, 32 + 42 = 52. 2 Fibonacci (13-th century) showed that 7 is a congruent number and conjectured that 1 is not a congruent number.

Anupam Saikia IIT Guwahati July 11, 2020 5 / 28 What Is a Congruent Number

Let Z = {0, ±1, ±2,...}, the set of integers, and

Congruent m Numbers Q = { n | m, n are inetgers, n 6= 0}, the set of rational numbers. Elliptic Curves A positive integer n is called congruent if it is the area of a right triangle all of which sides are rational numbers, i.e., 1 n = pq, p2 + q2 = r2, p, q, r ∈ . 2 Q

r   q p, q, r ∈ Q (a P ythagorean triple) 1  n = Area = 2 ×p×q p

Example: 6 is a congruent number as 1 6 = .3.4, 32 + 42 = 52. 2 Fibonacci (13-th century) showed that 7 is a congruent number and conjectured that 1 is not a congruent number.

Anupam Saikia IIT Guwahati July 11, 2020 5 / 28 What Is a Congruent Number

Let Z = {0, ±1, ±2,...}, the set of integers, and

Congruent m Numbers Q = { n | m, n are inetgers, n 6= 0}, the set of rational numbers. Elliptic Curves A positive integer n is called congruent if it is the area of a right triangle all of which sides are rational numbers, i.e., 1 n = pq, p2 + q2 = r2, p, q, r ∈ . 2 Q

r   q p, q, r ∈ Q (a P ythagorean triple) 1  n = Area = 2 ×p×q p

Example: 6 is a congruent number as 1 6 = .3.4, 32 + 42 = 52. 2 Fibonacci (13-th century) showed that 7 is a congruent number and conjectured that 1 is not a congruent number.

Anupam Saikia IIT Guwahati July 11, 2020 5 / 28 What Is a Congruent Number

Let Z = {0, ±1, ±2,...}, the set of integers, and

Congruent m Numbers Q = { n | m, n are inetgers, n 6= 0}, the set of rational numbers. Elliptic Curves A positive integer n is called congruent if it is the area of a right triangle all of which sides are rational numbers, i.e., 1 n = pq, p2 + q2 = r2, p, q, r ∈ . 2 Q

r   q p, q, r ∈ Q (a P ythagorean triple) 1  n = Area = 2 ×p×q p

Example: 6 is a congruent number as 1 6 = .3.4, 32 + 42 = 52. 2 Fibonacci (13-th century) showed that 7 is a congruent number and conjectured that 1 is not a congruent number.

Anupam Saikia IIT Guwahati July 11, 2020 5 / 28 What Is a Congruent Number

Let Z = {0, ±1, ±2,...}, the set of integers, and

Congruent m Numbers Q = { n | m, n are inetgers, n 6= 0}, the set of rational numbers. Elliptic Curves A positive integer n is called congruent if it is the area of a right triangle all of which sides are rational numbers, i.e., 1 n = pq, p2 + q2 = r2, p, q, r ∈ . 2 Q

r   q p, q, r ∈ Q (a P ythagorean triple) 1  n = Area = 2 ×p×q p

Example: 6 is a congruent number as 1 6 = .3.4, 32 + 42 = 52. 2 Fibonacci (13-th century) showed that 7 is a congruent number and conjectured that 1 is not a congruent number.

Anupam Saikia IIT Guwahati July 11, 2020 5 / 28 What Is a Congruent Number

Let Z = {0, ±1, ±2,...}, the set of integers, and

Congruent m Numbers Q = { n | m, n are inetgers, n 6= 0}, the set of rational numbers. Elliptic Curves A positive integer n is called congruent if it is the area of a right triangle all of which sides are rational numbers, i.e., 1 n = pq, p2 + q2 = r2, p, q, r ∈ . 2 Q

r   q p, q, r ∈ Q (a P ythagorean triple) 1  n = Area = 2 ×p×q p

Example: 6 is a congruent number as 1 6 = .3.4, 32 + 42 = 52. 2 Fibonacci (13-th century) showed that 7 is a congruent number and conjectured that 1 is not a congruent number.

Anupam Saikia IIT Guwahati July 11, 2020 5 / 28 What Is a Congruent Number

Let Z = {0, ±1, ±2,...}, the set of integers, and

Congruent m Numbers Q = { n | m, n are inetgers, n 6= 0}, the set of rational numbers. Elliptic Curves A positive integer n is called congruent if it is the area of a right triangle all of which sides are rational numbers, i.e., 1 n = pq, p2 + q2 = r2, p, q, r ∈ . 2 Q

r   q p, q, r ∈ Q (a P ythagorean triple) 1  n = Area = 2 ×p×q p

Example: 6 is a congruent number as 1 6 = .3.4, 32 + 42 = 52. 2 Fibonacci (13-th century) showed that 7 is a congruent number and conjectured that 1 is not a congruent number.

Anupam Saikia IIT Guwahati July 11, 2020 5 / 28 What Is a Congruent Number

Let Z = {0, ±1, ±2,...}, the set of integers, and

Congruent m Numbers Q = { n | m, n are inetgers, n 6= 0}, the set of rational numbers. Elliptic Curves A positive integer n is called congruent if it is the area of a right triangle all of which sides are rational numbers, i.e., 1 n = pq, p2 + q2 = r2, p, q, r ∈ . 2 Q

r   q p, q, r ∈ Q (a P ythagorean triple) 1  n = Area = 2 ×p×q p

Example: 6 is a congruent number as 1 6 = .3.4, 32 + 42 = 52. 2 Fibonacci (13-th century) showed that 7 is a congruent number and conjectured that 1 is not a congruent number.

Anupam Saikia IIT Guwahati July 11, 2020 5 / 28 Is 1 a Congruent Number?

Suppose 1 is the area of a right triangle with rational sides, so Congruent Numbers u  p 2 r 2 u 2 Elliptic Curves v  r ( q ) +( s ) =( v )  s  Area= 1 × p × r =1. p 2 q s q

Multiplying the sides by the product of the denominators qsv,  uqs  2 2 2  rqv (psv) +(rsv) =(uqs)  1 1 p r 2 2  Area= 2 ×psv×rqv= 2 × q × s ×(qsv) =(qsv) psv

If 1 were a congruent number, there will be a right-triangle with integer sides with a perfect square as area. Fermat showed that if such a triangle exists, then one can keep constructing smaller triangles with the same property, which leads to a contradiction.

Anupam Saikia IIT Guwahati July 11, 2020 6 / 28 Is 1 a Congruent Number?

Suppose 1 is the area of a right triangle with rational sides, so Congruent Numbers u  p 2 r 2 u 2 Elliptic Curves v  r ( q ) +( s ) =( v )  s  Area= 1 × p × r =1. p 2 q s q

Multiplying the sides by the product of the denominators qsv,  uqs  2 2 2  rqv (psv) +(rsv) =(uqs)  1 1 p r 2 2  Area= 2 ×psv×rqv= 2 × q × s ×(qsv) =(qsv) psv

If 1 were a congruent number, there will be a right-triangle with integer sides with a perfect square as area. Fermat showed that if such a triangle exists, then one can keep constructing smaller triangles with the same property, which leads to a contradiction.

Anupam Saikia IIT Guwahati July 11, 2020 6 / 28 Is 1 a Congruent Number?

Suppose 1 is the area of a right triangle with rational sides, so Congruent Numbers u  p 2 r 2 u 2 Elliptic Curves v  r ( q ) +( s ) =( v )  s  Area= 1 × p × r =1. p 2 q s q

Multiplying the sides by the product of the denominators qsv,  uqs  2 2 2  rqv (psv) +(rsv) =(uqs)  1 1 p r 2 2  Area= 2 ×psv×rqv= 2 × q × s ×(qsv) =(qsv) psv

If 1 were a congruent number, there will be a right-triangle with integer sides with a perfect square as area. Fermat showed that if such a triangle exists, then one can keep constructing smaller triangles with the same property, which leads to a contradiction.

Anupam Saikia IIT Guwahati July 11, 2020 6 / 28 Is 1 a Congruent Number?

Suppose 1 is the area of a right triangle with rational sides, so Congruent Numbers u  p 2 r 2 u 2 Elliptic Curves v  r ( q ) +( s ) =( v )  s  Area= 1 × p × r =1. p 2 q s q

Multiplying the sides by the product of the denominators qsv,  uqs  2 2 2  rqv (psv) +(rsv) =(uqs)  1 1 p r 2 2  Area= 2 ×psv×rqv= 2 × q × s ×(qsv) =(qsv) psv

If 1 were a congruent number, there will be a right-triangle with integer sides with a perfect square as area. Fermat showed that if such a triangle exists, then one can keep constructing smaller triangles with the same property, which leads to a contradiction.

Anupam Saikia IIT Guwahati July 11, 2020 6 / 28 Is 1 a Congruent Number?

Suppose 1 is the area of a right triangle with rational sides, so Congruent Numbers u  p 2 r 2 u 2 Elliptic Curves v  r ( q ) +( s ) =( v )  s  Area= 1 × p × r =1. p 2 q s q

Multiplying the sides by the product of the denominators qsv,  uqs  2 2 2  rqv (psv) +(rsv) =(uqs)  1 1 p r 2 2  Area= 2 ×psv×rqv= 2 × q × s ×(qsv) =(qsv) psv

If 1 were a congruent number, there will be a right-triangle with integer sides with a perfect square as area. Fermat showed that if such a triangle exists, then one can keep constructing smaller triangles with the same property, which leads to a contradiction.

Anupam Saikia IIT Guwahati July 11, 2020 6 / 28 Is 1 a Congruent Number?

Suppose 1 is the area of a right triangle with rational sides, so Congruent Numbers u  p 2 r 2 u 2 Elliptic Curves v  r ( q ) +( s ) =( v )  s  Area= 1 × p × r =1. p 2 q s q

Multiplying the sides by the product of the denominators qsv,  uqs  2 2 2  rqv (psv) +(rsv) =(uqs)  1 1 p r 2 2  Area= 2 ×psv×rqv= 2 × q × s ×(qsv) =(qsv) psv

If 1 were a congruent number, there will be a right-triangle with integer sides with a perfect square as area. Fermat showed that if such a triangle exists, then one can keep constructing smaller triangles with the same property, which leads to a contradiction.

Anupam Saikia IIT Guwahati July 11, 2020 6 / 28 Right Triangles with Integer Sides

Suppose a2 + b2 = c2 where a, b, c are coprime integers. We Congruent Numbers can assume that b is even and a and c are odd. Elliptic Curves Then c + a and c − a have greatest common divisor 2.

Now, b2 = c2 − a2 = (c + a)(c − a). Therefore, c + a = 2u2, c − a = 2v2 for some coprime integers u and v (by unique factorization in integers).

This gives a = u2 − v2, c = u2 + v2 and b = 2uv.

For example, we know that 52 + 122 = 132, and we have 5 = 32 − 22, 13 = 32 + 22 and 12 = 2 × 3 × 2.

This argument dates back to Euclid (300 BC).

Anupam Saikia IIT Guwahati July 11, 2020 7 / 28 Right Triangles with Integer Sides

Suppose a2 + b2 = c2 where a, b, c are coprime integers. We Congruent Numbers can assume that b is even and a and c are odd. Elliptic Curves Then c + a and c − a have greatest common divisor 2.

Now, b2 = c2 − a2 = (c + a)(c − a). Therefore, c + a = 2u2, c − a = 2v2 for some coprime integers u and v (by unique factorization in integers).

This gives a = u2 − v2, c = u2 + v2 and b = 2uv.

For example, we know that 52 + 122 = 132, and we have 5 = 32 − 22, 13 = 32 + 22 and 12 = 2 × 3 × 2.

This argument dates back to Euclid (300 BC).

Anupam Saikia IIT Guwahati July 11, 2020 7 / 28 Right Triangles with Integer Sides

Suppose a2 + b2 = c2 where a, b, c are coprime integers. We Congruent Numbers can assume that b is even and a and c are odd. Elliptic Curves Then c + a and c − a have greatest common divisor 2.

Now, b2 = c2 − a2 = (c + a)(c − a). Therefore, c + a = 2u2, c − a = 2v2 for some coprime integers u and v (by unique factorization in integers).

This gives a = u2 − v2, c = u2 + v2 and b = 2uv.

For example, we know that 52 + 122 = 132, and we have 5 = 32 − 22, 13 = 32 + 22 and 12 = 2 × 3 × 2.

This argument dates back to Euclid (300 BC).

Anupam Saikia IIT Guwahati July 11, 2020 7 / 28 Right Triangles with Integer Sides

Suppose a2 + b2 = c2 where a, b, c are coprime integers. We Congruent Numbers can assume that b is even and a and c are odd. Elliptic Curves Then c + a and c − a have greatest common divisor 2.

Now, b2 = c2 − a2 = (c + a)(c − a). Therefore, c + a = 2u2, c − a = 2v2 for some coprime integers u and v (by unique factorization in integers).

This gives a = u2 − v2, c = u2 + v2 and b = 2uv.

For example, we know that 52 + 122 = 132, and we have 5 = 32 − 22, 13 = 32 + 22 and 12 = 2 × 3 × 2.

This argument dates back to Euclid (300 BC).

Anupam Saikia IIT Guwahati July 11, 2020 7 / 28 Right Triangles with Integer Sides

Suppose a2 + b2 = c2 where a, b, c are coprime integers. We Congruent Numbers can assume that b is even and a and c are odd. Elliptic Curves Then c + a and c − a have greatest common divisor 2.

Now, b2 = c2 − a2 = (c + a)(c − a). Therefore, c + a = 2u2, c − a = 2v2 for some coprime integers u and v (by unique factorization in integers).

This gives a = u2 − v2, c = u2 + v2 and b = 2uv.

For example, we know that 52 + 122 = 132, and we have 5 = 32 − 22, 13 = 32 + 22 and 12 = 2 × 3 × 2.

This argument dates back to Euclid (300 BC).

Anupam Saikia IIT Guwahati July 11, 2020 7 / 28 Right Triangles with Integer Sides

Suppose a2 + b2 = c2 where a, b, c are coprime integers. We Congruent Numbers can assume that b is even and a and c are odd. Elliptic Curves Then c + a and c − a have greatest common divisor 2.

Now, b2 = c2 − a2 = (c + a)(c − a). Therefore, c + a = 2u2, c − a = 2v2 for some coprime integers u and v (by unique factorization in integers).

This gives a = u2 − v2, c = u2 + v2 and b = 2uv.

For example, we know that 52 + 122 = 132, and we have 5 = 32 − 22, 13 = 32 + 22 and 12 = 2 × 3 × 2.

This argument dates back to Euclid (300 BC).

Anupam Saikia IIT Guwahati July 11, 2020 7 / 28 Fermat’s Infinite Descent

 a, b, c ∈ Z c  b Euclid Congruent  2 2 2 2 Numbers =⇒ a = u −v , b = 2uv, c = u +v  gcd(a,b)=1 Elliptic Curves a gcd(u,v) = 1

If D2 is the area of a right-triangle as above, then 1 D2 = ab = uv(u + v)(u − v). 2

By coprimality, u = m2, v = n2, u + v = l2, u − v = k2.

2 2 2 2 2 2m (l+k) +(l−k) = 2(l +k ) = 2·(2u) = (2m)  l−k  Area = 1 (l+k)(l−k) = v = n2

Thus, if there exists a right-triangle with integer sides with a perfect square D2 as area, we can keep constructing such triangles with smaller integer sides (infinite descent). Therefore, 1 is not a congruent number. Anupam Saikia IIT Guwahati July 11, 2020 8 / 28 Fermat’s Infinite Descent

 a, b, c ∈ Z c  b Euclid Congruent  2 2 2 2 Numbers =⇒ a = u −v , b = 2uv, c = u +v  gcd(a,b)=1 Elliptic Curves a gcd(u,v) = 1

If D2 is the area of a right-triangle as above, then 1 D2 = ab = uv(u + v)(u − v). 2

By coprimality, u = m2, v = n2, u + v = l2, u − v = k2.

2 2 2 2 2 2m (l+k) +(l−k) = 2(l +k ) = 2·(2u) = (2m)  l−k  Area = 1 (l+k)(l−k) = v = n2

Thus, if there exists a right-triangle with integer sides with a perfect square D2 as area, we can keep constructing such triangles with smaller integer sides (infinite descent). Therefore, 1 is not a congruent number. Anupam Saikia IIT Guwahati July 11, 2020 8 / 28 Fermat’s Infinite Descent

 a, b, c ∈ Z c  b Euclid Congruent  2 2 2 2 Numbers =⇒ a = u −v , b = 2uv, c = u +v  gcd(a,b)=1 Elliptic Curves a gcd(u,v) = 1

If D2 is the area of a right-triangle as above, then 1 D2 = ab = uv(u + v)(u − v). 2

By coprimality, u = m2, v = n2, u + v = l2, u − v = k2.

2 2 2 2 2 2m (l+k) +(l−k) = 2(l +k ) = 2·(2u) = (2m)  l−k  Area = 1 (l+k)(l−k) = v = n2

Thus, if there exists a right-triangle with integer sides with a perfect square D2 as area, we can keep constructing such triangles with smaller integer sides (infinite descent). Therefore, 1 is not a congruent number. Anupam Saikia IIT Guwahati July 11, 2020 8 / 28 Fermat’s Infinite Descent

 a, b, c ∈ Z c  b Euclid Congruent  2 2 2 2 Numbers =⇒ a = u −v , b = 2uv, c = u +v  gcd(a,b)=1 Elliptic Curves a gcd(u,v) = 1

If D2 is the area of a right-triangle as above, then 1 D2 = ab = uv(u + v)(u − v). 2

By coprimality, u = m2, v = n2, u + v = l2, u − v = k2.

2 2 2 2 2 2m (l+k) +(l−k) = 2(l +k ) = 2·(2u) = (2m)  l−k  Area = 1 (l+k)(l−k) = v = n2

Thus, if there exists a right-triangle with integer sides with a perfect square D2 as area, we can keep constructing such triangles with smaller integer sides (infinite descent). Therefore, 1 is not a congruent number. Anupam Saikia IIT Guwahati July 11, 2020 8 / 28 Fermat’s Infinite Descent

 a, b, c ∈ Z c  b Euclid Congruent  2 2 2 2 Numbers =⇒ a = u −v , b = 2uv, c = u +v  gcd(a,b)=1 Elliptic Curves a gcd(u,v) = 1

If D2 is the area of a right-triangle as above, then 1 D2 = ab = uv(u + v)(u − v). 2

By coprimality, u = m2, v = n2, u + v = l2, u − v = k2.

2 2 2 2 2 2m (l+k) +(l−k) = 2(l +k ) = 2·(2u) = (2m)  l−k  Area = 1 (l+k)(l−k) = v = n2

Thus, if there exists a right-triangle with integer sides with a perfect square D2 as area, we can keep constructing such triangles with smaller integer sides (infinite descent). Therefore, 1 is not a congruent number. Anupam Saikia IIT Guwahati July 11, 2020 8 / 28 Fermat’s Infinite Descent

 a, b, c ∈ Z c  b Euclid Congruent  2 2 2 2 Numbers =⇒ a = u −v , b = 2uv, c = u +v  gcd(a,b)=1 Elliptic Curves a gcd(u,v) = 1

If D2 is the area of a right-triangle as above, then 1 D2 = ab = uv(u + v)(u − v). 2

By coprimality, u = m2, v = n2, u + v = l2, u − v = k2.

2 2 2 2 2 2m (l+k) +(l−k) = 2(l +k ) = 2·(2u) = (2m)  l−k  Area = 1 (l+k)(l−k) = v = n2

Thus, if there exists a right-triangle with integer sides with a perfect square D2 as area, we can keep constructing such triangles with smaller integer sides (infinite descent). Therefore, 1 is not a congruent number. Anupam Saikia IIT Guwahati July 11, 2020 8 / 28 Fermat’s Infinite Descent

 a, b, c ∈ Z c  b Euclid Congruent  2 2 2 2 Numbers =⇒ a = u −v , b = 2uv, c = u +v  gcd(a,b)=1 Elliptic Curves a gcd(u,v) = 1

If D2 is the area of a right-triangle as above, then 1 D2 = ab = uv(u + v)(u − v). 2

By coprimality, u = m2, v = n2, u + v = l2, u − v = k2.

2 2 2 2 2 2m (l+k) +(l−k) = 2(l +k ) = 2·(2u) = (2m)  l−k  Area = 1 (l+k)(l−k) = v = n2

Thus, if there exists a right-triangle with integer sides with a perfect square D2 as area, we can keep constructing such triangles with smaller integer sides (infinite descent). Therefore, 1 is not a congruent number. Anupam Saikia IIT Guwahati July 11, 2020 8 / 28 Fermat’s Infinite Descent

 a, b, c ∈ Z c  b Euclid Congruent  2 2 2 2 Numbers =⇒ a = u −v , b = 2uv, c = u +v  gcd(a,b)=1 Elliptic Curves a gcd(u,v) = 1

If D2 is the area of a right-triangle as above, then 1 D2 = ab = uv(u + v)(u − v). 2

By coprimality, u = m2, v = n2, u + v = l2, u − v = k2.

2 2 2 2 2 2m (l+k) +(l−k) = 2(l +k ) = 2·(2u) = (2m)  l−k  Area = 1 (l+k)(l−k) = v = n2

Thus, if there exists a right-triangle with integer sides with a perfect square D2 as area, we can keep constructing such triangles with smaller integer sides (infinite descent). Therefore, 1 is not a congruent number. Anupam Saikia IIT Guwahati July 11, 2020 8 / 28 Fermat’s Infinite Descent

 a, b, c ∈ Z c  b Euclid Congruent  2 2 2 2 Numbers =⇒ a = u −v , b = 2uv, c = u +v  gcd(a,b)=1 Elliptic Curves a gcd(u,v) = 1

If D2 is the area of a right-triangle as above, then 1 D2 = ab = uv(u + v)(u − v). 2

By coprimality, u = m2, v = n2, u + v = l2, u − v = k2.

2 2 2 2 2 2m (l+k) +(l−k) = 2(l +k ) = 2·(2u) = (2m)  l−k  Area = 1 (l+k)(l−k) = v = n2

Thus, if there exists a right-triangle with integer sides with a perfect square D2 as area, we can keep constructing such triangles with smaller integer sides (infinite descent). Therefore, 1 is not a congruent number. Anupam Saikia IIT Guwahati July 11, 2020 8 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 Reformulation

5 is a congruent number as it is the area of a right triangle Congruent Numbers 3 20 41 with sides 2 , 3 and 6 . We can see that it may not be easy Elliptic Curves to find the sides directly, and a reformulation indeed helps.

One can relate a rational right triangle of area n to a rational 2 3 2 point (x, y) on the curve En : y = x − n x as follows:

r 2 r(p−q)(p+q) Given (p,q,r), take x = ( 2 ) , y = 8 =⇒ y2 = x3−n2x.  p, q, r ∈ Q r  Conversely, for 06=y, x, ∈ , y2 = x3−n2x, q Q  2 2 2 2 1 x −n 2xn x +n  Area= ·p·q=n p = | y |, q = | y |, r = | y | ∈ Q p 2 2 2 2 1 =⇒ p +q = r , 2 pq = n

Anupam Saikia IIT Guwahati July 11, 2020 9 / 28 An Example

Example: Find a right triangle of rational sides with 5 as area. Congruent Numbers Equivalently, find a point on y2 = x3 − 52x with non-zero Elliptic Curves y-coordinate.

The point S = (−4, 6) clearly lies on y2 = x3 − 52x.

Then S corresponds to a right triangle with sides

2 2 x2−n2 (−4) −5 3 2xn 2.(−4).5 20 x2+n2 41 p=| y |=| 6 |= 2 , q=| y |=| 6 |= 3 , r=| y |= 6 ,

1 3 20 with area 2 . 2 . 3 = 5.

The question whether n is a congruent number reduces to the question of finding a rational point (x, y) y 6= 0 on the curve 2 3 2 En : y = x − n x.

En belongs to a family of curves known as Elliptic Curves.

Anupam Saikia IIT Guwahati July 11, 2020 10 / 28 An Example

Example: Find a right triangle of rational sides with 5 as area. Congruent Numbers Equivalently, find a point on y2 = x3 − 52x with non-zero Elliptic Curves y-coordinate.

The point S = (−4, 6) clearly lies on y2 = x3 − 52x.

Then S corresponds to a right triangle with sides

2 2 x2−n2 (−4) −5 3 2xn 2.(−4).5 20 x2+n2 41 p=| y |=| 6 |= 2 , q=| y |=| 6 |= 3 , r=| y |= 6 ,

1 3 20 with area 2 . 2 . 3 = 5.

The question whether n is a congruent number reduces to the question of finding a rational point (x, y) y 6= 0 on the curve 2 3 2 En : y = x − n x.

En belongs to a family of curves known as Elliptic Curves.

Anupam Saikia IIT Guwahati July 11, 2020 10 / 28 An Example

Example: Find a right triangle of rational sides with 5 as area. Congruent Numbers Equivalently, find a point on y2 = x3 − 52x with non-zero Elliptic Curves y-coordinate.

The point S = (−4, 6) clearly lies on y2 = x3 − 52x.

Then S corresponds to a right triangle with sides

2 2 x2−n2 (−4) −5 3 2xn 2.(−4).5 20 x2+n2 41 p=| y |=| 6 |= 2 , q=| y |=| 6 |= 3 , r=| y |= 6 ,

1 3 20 with area 2 . 2 . 3 = 5.

The question whether n is a congruent number reduces to the question of finding a rational point (x, y) y 6= 0 on the curve 2 3 2 En : y = x − n x.

En belongs to a family of curves known as Elliptic Curves.

Anupam Saikia IIT Guwahati July 11, 2020 10 / 28 An Example

Example: Find a right triangle of rational sides with 5 as area. Congruent Numbers Equivalently, find a point on y2 = x3 − 52x with non-zero Elliptic Curves y-coordinate.

The point S = (−4, 6) clearly lies on y2 = x3 − 52x.

Then S corresponds to a right triangle with sides

2 2 x2−n2 (−4) −5 3 2xn 2.(−4).5 20 x2+n2 41 p=| y |=| 6 |= 2 , q=| y |=| 6 |= 3 , r=| y |= 6 ,

1 3 20 with area 2 . 2 . 3 = 5.

The question whether n is a congruent number reduces to the question of finding a rational point (x, y) y 6= 0 on the curve 2 3 2 En : y = x − n x.

En belongs to a family of curves known as Elliptic Curves.

Anupam Saikia IIT Guwahati July 11, 2020 10 / 28 An Example

Example: Find a right triangle of rational sides with 5 as area. Congruent Numbers Equivalently, find a point on y2 = x3 − 52x with non-zero Elliptic Curves y-coordinate.

The point S = (−4, 6) clearly lies on y2 = x3 − 52x.

Then S corresponds to a right triangle with sides

2 2 x2−n2 (−4) −5 3 2xn 2.(−4).5 20 x2+n2 41 p=| y |=| 6 |= 2 , q=| y |=| 6 |= 3 , r=| y |= 6 ,

1 3 20 with area 2 . 2 . 3 = 5.

The question whether n is a congruent number reduces to the question of finding a rational point (x, y) y 6= 0 on the curve 2 3 2 En : y = x − n x.

En belongs to a family of curves known as Elliptic Curves.

Anupam Saikia IIT Guwahati July 11, 2020 10 / 28 An Example

Example: Find a right triangle of rational sides with 5 as area. Congruent Numbers Equivalently, find a point on y2 = x3 − 52x with non-zero Elliptic Curves y-coordinate.

The point S = (−4, 6) clearly lies on y2 = x3 − 52x.

Then S corresponds to a right triangle with sides

2 2 x2−n2 (−4) −5 3 2xn 2.(−4).5 20 x2+n2 41 p=| y |=| 6 |= 2 , q=| y |=| 6 |= 3 , r=| y |= 6 ,

1 3 20 with area 2 . 2 . 3 = 5.

The question whether n is a congruent number reduces to the question of finding a rational point (x, y) y 6= 0 on the curve 2 3 2 En : y = x − n x.

En belongs to a family of curves known as Elliptic Curves.

Anupam Saikia IIT Guwahati July 11, 2020 10 / 28 An Example

Example: Find a right triangle of rational sides with 5 as area. Congruent Numbers Equivalently, find a point on y2 = x3 − 52x with non-zero Elliptic Curves y-coordinate.

The point S = (−4, 6) clearly lies on y2 = x3 − 52x.

Then S corresponds to a right triangle with sides

2 2 x2−n2 (−4) −5 3 2xn 2.(−4).5 20 x2+n2 41 p=| y |=| 6 |= 2 , q=| y |=| 6 |= 3 , r=| y |= 6 ,

1 3 20 with area 2 . 2 . 3 = 5.

The question whether n is a congruent number reduces to the question of finding a rational point (x, y) y 6= 0 on the curve 2 3 2 En : y = x − n x.

En belongs to a family of curves known as Elliptic Curves.

Anupam Saikia IIT Guwahati July 11, 2020 10 / 28 An Example

Example: Find a right triangle of rational sides with 5 as area. Congruent Numbers Equivalently, find a point on y2 = x3 − 52x with non-zero Elliptic Curves y-coordinate.

The point S = (−4, 6) clearly lies on y2 = x3 − 52x.

Then S corresponds to a right triangle with sides

2 2 x2−n2 (−4) −5 3 2xn 2.(−4).5 20 x2+n2 41 p=| y |=| 6 |= 2 , q=| y |=| 6 |= 3 , r=| y |= 6 ,

1 3 20 with area 2 . 2 . 3 = 5.

The question whether n is a congruent number reduces to the question of finding a rational point (x, y) y 6= 0 on the curve 2 3 2 En : y = x − n x.

En belongs to a family of curves known as Elliptic Curves.

Anupam Saikia IIT Guwahati July 11, 2020 10 / 28 Sections

Congruent Numbers

Elliptic Curves

Anupam Saikia IIT Guwahati July 11, 2020 11 / 28 Sections

Congruent Numbers

Elliptic Curves

1 Congruent Numbers

2 Elliptic Curves

Anupam Saikia IIT Guwahati July 11, 2020 12 / 28 Elliptic Curves

Elliptic curves are ubiquitous in mathematics. Congruent Numbers Study of elliptic curves brings together number theory, algebra, Elliptic Curves analysis and algebraic geometry.

Elliptic curves have been used to prove Fermat’s Last Theorem, one of the most remarkable results in mathematics.

Elliptic curves have provided breakthrough toward resolving the Congruent Number Problem.

One of the seven millennium problems, the Birch and Swinnerton-Dyer Conjecture, is a prediction about relation between the algebraic and the analytic properties of an elliptic curve.

Elliptic curves have been very useful in cryptography, i.e., in coding messages for security.

Anupam Saikia IIT Guwahati July 11, 2020 13 / 28 Elliptic Curves

Elliptic curves are ubiquitous in mathematics. Congruent Numbers Study of elliptic curves brings together number theory, algebra, Elliptic Curves analysis and algebraic geometry.

Elliptic curves have been used to prove Fermat’s Last Theorem, one of the most remarkable results in mathematics.

Elliptic curves have provided breakthrough toward resolving the Congruent Number Problem.

One of the seven millennium problems, the Birch and Swinnerton-Dyer Conjecture, is a prediction about relation between the algebraic and the analytic properties of an elliptic curve.

Elliptic curves have been very useful in cryptography, i.e., in coding messages for security.

Anupam Saikia IIT Guwahati July 11, 2020 13 / 28 Elliptic Curves

Elliptic curves are ubiquitous in mathematics. Congruent Numbers Study of elliptic curves brings together number theory, algebra, Elliptic Curves analysis and algebraic geometry.

Elliptic curves have been used to prove Fermat’s Last Theorem, one of the most remarkable results in mathematics.

Elliptic curves have provided breakthrough toward resolving the Congruent Number Problem.

One of the seven millennium problems, the Birch and Swinnerton-Dyer Conjecture, is a prediction about relation between the algebraic and the analytic properties of an elliptic curve.

Elliptic curves have been very useful in cryptography, i.e., in coding messages for security.

Anupam Saikia IIT Guwahati July 11, 2020 13 / 28 Elliptic Curves

Elliptic curves are ubiquitous in mathematics. Congruent Numbers Study of elliptic curves brings together number theory, algebra, Elliptic Curves analysis and algebraic geometry.

Elliptic curves have been used to prove Fermat’s Last Theorem, one of the most remarkable results in mathematics.

Elliptic curves have provided breakthrough toward resolving the Congruent Number Problem.

One of the seven millennium problems, the Birch and Swinnerton-Dyer Conjecture, is a prediction about relation between the algebraic and the analytic properties of an elliptic curve.

Elliptic curves have been very useful in cryptography, i.e., in coding messages for security.

Anupam Saikia IIT Guwahati July 11, 2020 13 / 28 Elliptic Curves

Elliptic curves are ubiquitous in mathematics. Congruent Numbers Study of elliptic curves brings together number theory, algebra, Elliptic Curves analysis and algebraic geometry.

Elliptic curves have been used to prove Fermat’s Last Theorem, one of the most remarkable results in mathematics.

Elliptic curves have provided breakthrough toward resolving the Congruent Number Problem.

One of the seven millennium problems, the Birch and Swinnerton-Dyer Conjecture, is a prediction about relation between the algebraic and the analytic properties of an elliptic curve.

Elliptic curves have been very useful in cryptography, i.e., in coding messages for security.

Anupam Saikia IIT Guwahati July 11, 2020 13 / 28 Elliptic Curves

Elliptic curves are ubiquitous in mathematics. Congruent Numbers Study of elliptic curves brings together number theory, algebra, Elliptic Curves analysis and algebraic geometry.

Elliptic curves have been used to prove Fermat’s Last Theorem, one of the most remarkable results in mathematics.

Elliptic curves have provided breakthrough toward resolving the Congruent Number Problem.

One of the seven millennium problems, the Birch and Swinnerton-Dyer Conjecture, is a prediction about relation between the algebraic and the analytic properties of an elliptic curve.

Elliptic curves have been very useful in cryptography, i.e., in coding messages for security.

Anupam Saikia IIT Guwahati July 11, 2020 13 / 28 What is an Elliptic Curve

Elliptic curves probably appeared first in the book “Arithmetica” by Congruent the Greek mathematician Diophantus in the third century A.D.. Numbers

Elliptic Curves

An elliptic curve over a field F is a ‘non-singular curve’ defined by an equation of the form

y2 = x3 + ax + b, a, b ∈ F.

We usually take F as the field Q of rational numbers or a finite field ¯ ¯ ¯ ¯ ¯ Fp with p elements for a prime p. For example, F5 = {0, 1, 2, 3, 4}.

Anupam Saikia IIT Guwahati July 11, 2020 14 / 28 What is an Elliptic Curve

Elliptic curves probably appeared first in the book “Arithmetica” by Congruent the Greek mathematician Diophantus in the third century A.D.. Numbers

Elliptic Curves

An elliptic curve over a field F is a ‘non-singular curve’ defined by an equation of the form

y2 = x3 + ax + b, a, b ∈ F.

We usually take F as the field Q of rational numbers or a finite field ¯ ¯ ¯ ¯ ¯ Fp with p elements for a prime p. For example, F5 = {0, 1, 2, 3, 4}.

Anupam Saikia IIT Guwahati July 11, 2020 14 / 28 What is an Elliptic Curve

Elliptic curves probably appeared first in the book “Arithmetica” by Congruent the Greek mathematician Diophantus in the third century A.D.. Numbers

Elliptic Curves

An elliptic curve over a field F is a ‘non-singular curve’ defined by an equation of the form

y2 = x3 + ax + b, a, b ∈ F.

We usually take F as the field Q of rational numbers or a finite field ¯ ¯ ¯ ¯ ¯ Fp with p elements for a prime p. For example, F5 = {0, 1, 2, 3, 4}.

Anupam Saikia IIT Guwahati July 11, 2020 14 / 28 What is an Elliptic Curve

Elliptic curves probably appeared first in the book “Arithmetica” by Congruent the Greek mathematician Diophantus in the third century A.D.. Numbers

Elliptic Curves

An elliptic curve over a field F is a ‘non-singular curve’ defined by an equation of the form

y2 = x3 + ax + b, a, b ∈ F.

We usually take F as the field Q of rational numbers or a finite field ¯ ¯ ¯ ¯ ¯ Fp with p elements for a prime p. For example, F5 = {0, 1, 2, 3, 4}.

Anupam Saikia IIT Guwahati July 11, 2020 14 / 28 Speciality of Elliptic Curves

Elliptic curves have certain properties that distinguish them from

Congruent all other curves. Numbers Elliptic Curves An elliptic curve can contain infinitely many points with rational numbers as co-ordinates.

Two points on an elliptic curve can be ‘added’ to get a third point on the curve. Further, this operation of adding points is commutative, associative, has an identity and each point has an inverse too. In other words, the points on an elliptic curve form an ‘abelian group’. For example, the set of all integers Z forms an abelian group under usual addition, as n + m = m + n ∈ Z, m + 0 = m, l + (m + n) = (l + m) + n and m + (−m) = 0 for any integers m, n and l.

Anupam Saikia IIT Guwahati July 11, 2020 15 / 28 Speciality of Elliptic Curves

Elliptic curves have certain properties that distinguish them from

Congruent all other curves. Numbers Elliptic Curves An elliptic curve can contain infinitely many points with rational numbers as co-ordinates.

Two points on an elliptic curve can be ‘added’ to get a third point on the curve. Further, this operation of adding points is commutative, associative, has an identity and each point has an inverse too. In other words, the points on an elliptic curve form an ‘abelian group’. For example, the set of all integers Z forms an abelian group under usual addition, as n + m = m + n ∈ Z, m + 0 = m, l + (m + n) = (l + m) + n and m + (−m) = 0 for any integers m, n and l.

Anupam Saikia IIT Guwahati July 11, 2020 15 / 28 Speciality of Elliptic Curves

Elliptic curves have certain properties that distinguish them from

Congruent all other curves. Numbers Elliptic Curves An elliptic curve can contain infinitely many points with rational numbers as co-ordinates.

Two points on an elliptic curve can be ‘added’ to get a third point on the curve. Further, this operation of adding points is commutative, associative, has an identity and each point has an inverse too. In other words, the points on an elliptic curve form an ‘abelian group’. For example, the set of all integers Z forms an abelian group under usual addition, as n + m = m + n ∈ Z, m + 0 = m, l + (m + n) = (l + m) + n and m + (−m) = 0 for any integers m, n and l.

Anupam Saikia IIT Guwahati July 11, 2020 15 / 28 Speciality of Elliptic Curves

Elliptic curves have certain properties that distinguish them from

Congruent all other curves. Numbers Elliptic Curves An elliptic curve can contain infinitely many points with rational numbers as co-ordinates.

Two points on an elliptic curve can be ‘added’ to get a third point on the curve. Further, this operation of adding points is commutative, associative, has an identity and each point has an inverse too. In other words, the points on an elliptic curve form an ‘abelian group’. For example, the set of all integers Z forms an abelian group under usual addition, as n + m = m + n ∈ Z, m + 0 = m, l + (m + n) = (l + m) + n and m + (−m) = 0 for any integers m, n and l.

Anupam Saikia IIT Guwahati July 11, 2020 15 / 28 Speciality of Elliptic Curves

Elliptic curves have certain properties that distinguish them from

Congruent all other curves. Numbers Elliptic Curves An elliptic curve can contain infinitely many points with rational numbers as co-ordinates.

Two points on an elliptic curve can be ‘added’ to get a third point on the curve. Further, this operation of adding points is commutative, associative, has an identity and each point has an inverse too. In other words, the points on an elliptic curve form an ‘abelian group’. For example, the set of all integers Z forms an abelian group under usual addition, as n + m = m + n ∈ Z, m + 0 = m, l + (m + n) = (l + m) + n and m + (−m) = 0 for any integers m, n and l.

Anupam Saikia IIT Guwahati July 11, 2020 15 / 28 Speciality of Elliptic Curves

Elliptic curves have certain properties that distinguish them from

Congruent all other curves. Numbers Elliptic Curves An elliptic curve can contain infinitely many points with rational numbers as co-ordinates.

Two points on an elliptic curve can be ‘added’ to get a third point on the curve. Further, this operation of adding points is commutative, associative, has an identity and each point has an inverse too. In other words, the points on an elliptic curve form an ‘abelian group’. For example, the set of all integers Z forms an abelian group under usual addition, as n + m = m + n ∈ Z, m + 0 = m, l + (m + n) = (l + m) + n and m + (−m) = 0 for any integers m, n and l.

Anupam Saikia IIT Guwahati July 11, 2020 15 / 28 The Point at Infinity on an Elliptic Curve

A non-vertical line will have three real points of intersection or one Congruent Numbers real and two complex points of intersection, which is also clear form Elliptic Curves the substitution y = mx + c in y2 = x3 + ax + b.

However, the vertical lines x = x0 will have either two real or two complex points of intersection. For a consistent theory, we need a third point of intersection. We adjoin an additional ‘point at infinity’ to the curve.

This point can be visualized as lying on the top (and the bottom) of the xy-plane at infinity.

Any two vertical lines intersect at the point at infinity, which we

denote by P∞.

Anupam Saikia IIT Guwahati July 11, 2020 16 / 28 The Point at Infinity on an Elliptic Curve

A non-vertical line will have three real points of intersection or one Congruent Numbers real and two complex points of intersection, which is also clear form Elliptic Curves the substitution y = mx + c in y2 = x3 + ax + b.

However, the vertical lines x = x0 will have either two real or two complex points of intersection. For a consistent theory, we need a third point of intersection. We adjoin an additional ‘point at infinity’ to the curve.

This point can be visualized as lying on the top (and the bottom) of the xy-plane at infinity.

Any two vertical lines intersect at the point at infinity, which we

denote by P∞.

Anupam Saikia IIT Guwahati July 11, 2020 16 / 28 The Point at Infinity on an Elliptic Curve

A non-vertical line will have three real points of intersection or one Congruent Numbers real and two complex points of intersection, which is also clear form Elliptic Curves the substitution y = mx + c in y2 = x3 + ax + b.

However, the vertical lines x = x0 will have either two real or two complex points of intersection. For a consistent theory, we need a third point of intersection. We adjoin an additional ‘point at infinity’ to the curve.

This point can be visualized as lying on the top (and the bottom) of the xy-plane at infinity.

Any two vertical lines intersect at the point at infinity, which we

denote by P∞.

Anupam Saikia IIT Guwahati July 11, 2020 16 / 28 The Point at Infinity on an Elliptic Curve

A non-vertical line will have three real points of intersection or one Congruent Numbers real and two complex points of intersection, which is also clear form Elliptic Curves the substitution y = mx + c in y2 = x3 + ax + b.

However, the vertical lines x = x0 will have either two real or two complex points of intersection. For a consistent theory, we need a third point of intersection. We adjoin an additional ‘point at infinity’ to the curve.

This point can be visualized as lying on the top (and the bottom) of the xy-plane at infinity.

Any two vertical lines intersect at the point at infinity, which we

denote by P∞.

Anupam Saikia IIT Guwahati July 11, 2020 16 / 28 The Point at Infinity on an Elliptic Curve

A non-vertical line will have three real points of intersection or one Congruent Numbers real and two complex points of intersection, which is also clear form Elliptic Curves the substitution y = mx + c in y2 = x3 + ax + b.

However, the vertical lines x = x0 will have either two real or two complex points of intersection. For a consistent theory, we need a third point of intersection. We adjoin an additional ‘point at infinity’ to the curve.

This point can be visualized as lying on the top (and the bottom) of the xy-plane at infinity.

Any two vertical lines intersect at the point at infinity, which we

denote by P∞.

Anupam Saikia IIT Guwahati July 11, 2020 16 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 Addition on Elliptic Curve: A Diagram

P∞: the point at infinity

Congruent Numbers 2 3 Elliptic Curves y=mx+c  y =x +ax+b  P ∗Q=(x3,y3)  ⇒(mx+c)2=x3+ax+b   Q=(x2,y2) 2  ⇒x1+x2+x3=m (x1,y1)=P 2 P ⊕Q=(x3,−y3) ⇒x3=m −x1−x2.

y2=x3+ax+b

Any two points P and Q on E can be added to obtain a third point P ⊕ Q on E. For this addition of points, we have

an identity, which is P∞, an inverse for each point, associativity and commutativity.

Anupam Saikia IIT Guwahati July 11, 2020 17 / 28 P∞ is the Identity

Congruent Numbers P∞: the point at infinity Elliptic Curves

P =(x1,y1)

P ∗P∞=(x1,−y1)

y2=x3+ax+b

The point P∞ serves as the identity for addition on elliptic curve.

Anupam Saikia IIT Guwahati July 11, 2020 18 / 28 P∞ is the Identity

Congruent Numbers P∞: the point at infinity Elliptic Curves

P =(x1,y1)

P ∗P∞=(x1,−y1)

y2=x3+ax+b

The point P∞ serves as the identity for addition on elliptic curve.

Anupam Saikia IIT Guwahati July 11, 2020 18 / 28 P∞ is the Identity

Congruent Numbers P∞: the point at infinity Elliptic Curves

P =(x1,y1)

P ∗P∞=(x1,−y1)

y2=x3+ax+b

The point P∞ serves as the identity for addition on elliptic curve.

Anupam Saikia IIT Guwahati July 11, 2020 18 / 28 P∞ is the Identity

Congruent Numbers P∞: the point at infinity Elliptic Curves

P =(x1,y1)

P ∗P∞=(x1,−y1)

y2=x3+ax+b

The point P∞ serves as the identity for addition on elliptic curve.

Anupam Saikia IIT Guwahati July 11, 2020 18 / 28 P∞ is the Identity

Congruent Numbers P∞: the point at infinity Elliptic Curves

P =(x1,y1)

P ∗P∞=(x1,−y1)

y2=x3+ax+b

The point P∞ serves as the identity for addition on elliptic curve.

Anupam Saikia IIT Guwahati July 11, 2020 18 / 28 The Inverse of a Point

P∞: the point at infinity

Congruent Numbers

Elliptic Curves P =(x1,y1)

P =(x1,−y1)

y2=x3+ax+b

The additive inverse of the point P = (x1, y1) is given by

P = (x1, −y1).

Anupam Saikia IIT Guwahati July 11, 2020 19 / 28 The Inverse of a Point

P∞: the point at infinity

Congruent Numbers

Elliptic Curves P =(x1,y1)

P =(x1,−y1)

y2=x3+ax+b

The additive inverse of the point P = (x1, y1) is given by

P = (x1, −y1).

Anupam Saikia IIT Guwahati July 11, 2020 19 / 28 The Inverse of a Point

P∞: the point at infinity

Congruent Numbers

Elliptic Curves P =(x1,y1)

P =(x1,−y1)

y2=x3+ax+b

The additive inverse of the point P = (x1, y1) is given by

P = (x1, −y1).

Anupam Saikia IIT Guwahati July 11, 2020 19 / 28 The Inverse of a Point

P∞: the point at infinity

Congruent Numbers

Elliptic Curves P =(x1,y1)

P =(x1,−y1)

y2=x3+ax+b

The additive inverse of the point P = (x1, y1) is given by

P = (x1, −y1).

Anupam Saikia IIT Guwahati July 11, 2020 19 / 28 The Inverse of a Point

P∞: the point at infinity

Congruent Numbers

Elliptic Curves P =(x1,y1)

P =(x1,−y1)

y2=x3+ax+b

The additive inverse of the point P = (x1, y1) is given by

P = (x1, −y1).

Anupam Saikia IIT Guwahati July 11, 2020 19 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Doubling of a Point

Congruent Numbers

Elliptic Curves P∞: the point at infinity

b (tangent at P ) b b 2 3 b P ⊕P =(x3,−y3) y =x +ax+b b y=mx+c (x1,y1)=P b ⇒(mx+c)2=x3+ax+b b b 2 b ⇒x1+x1+x3=m . 2 b ⇒x3=m −2x1. b P ∗P =(x ,y ) b 3 3 2 3 b y =x +ax+b b

The double of the point P = (x, y) is given by 2P = P ⊕ P = (x3, −y3).

Anupam Saikia IIT Guwahati July 11, 2020 20 / 28 Multiplication by Integers

We can add P to itself and get another point 2P = P ⊕ P , Congruent Numbers and in general, Elliptic Curves nP = P ⊕ P ⊕ ... ⊕ P , n = 1, 2, 3,.... | {z } n-times

For a negative integer, say n = −10, we can define (−10)P as (−10)P = [10P ],

where R denoted the inverse of a point R on the elliptic curve.

This way, we can define a multiplication on E by all integers.

Anupam Saikia IIT Guwahati July 11, 2020 21 / 28 Multiplication by Integers

We can add P to itself and get another point 2P = P ⊕ P , Congruent Numbers and in general, Elliptic Curves nP = P ⊕ P ⊕ ... ⊕ P , n = 1, 2, 3,.... | {z } n-times

For a negative integer, say n = −10, we can define (−10)P as (−10)P = [10P ],

where R denoted the inverse of a point R on the elliptic curve.

This way, we can define a multiplication on E by all integers.

Anupam Saikia IIT Guwahati July 11, 2020 21 / 28 Multiplication by Integers

We can add P to itself and get another point 2P = P ⊕ P , Congruent Numbers and in general, Elliptic Curves nP = P ⊕ P ⊕ ... ⊕ P , n = 1, 2, 3,.... | {z } n-times

For a negative integer, say n = −10, we can define (−10)P as (−10)P = [10P ],

where R denoted the inverse of a point R on the elliptic curve.

This way, we can define a multiplication on E by all integers.

Anupam Saikia IIT Guwahati July 11, 2020 21 / 28 Multiplication by Integers

We can add P to itself and get another point 2P = P ⊕ P , Congruent Numbers and in general, Elliptic Curves nP = P ⊕ P ⊕ ... ⊕ P , n = 1, 2, 3,.... | {z } n-times

For a negative integer, say n = −10, we can define (−10)P as (−10)P = [10P ],

where R denoted the inverse of a point R on the elliptic curve.

This way, we can define a multiplication on E by all integers.

Anupam Saikia IIT Guwahati July 11, 2020 21 / 28 The Mordell-Weil Group

2 3 Let E(Q) = {(x, y) | x, y ∈ Q, y = x + ax + b}. Then Congruent Numbers

Elliptic Curves P,Q ∈ E(Q) =⇒ P ⊕ Q ∈ E(Q).

The abelian group E(Q) is called the Mordell-Weil group of E/Q.

If the set S = {P, 2P, 3P, ···} is finite, then nP = P for some integer n. Such a point P is called a torsion point, and we denote

the subgroup forme by the torsion points on E(Q) as E(Q)tor.

Anupam Saikia IIT Guwahati July 11, 2020 22 / 28 The Mordell-Weil Group

2 3 Let E(Q) = {(x, y) | x, y ∈ Q, y = x + ax + b}. Then Congruent Numbers

Elliptic Curves P,Q ∈ E(Q) =⇒ P ⊕ Q ∈ E(Q).

The abelian group E(Q) is called the Mordell-Weil group of E/Q.

If the set S = {P, 2P, 3P, ···} is finite, then nP = P for some integer n. Such a point P is called a torsion point, and we denote

the subgroup forme by the torsion points on E(Q) as E(Q)tor.

Anupam Saikia IIT Guwahati July 11, 2020 22 / 28 The Mordell-Weil Group

2 3 Let E(Q) = {(x, y) | x, y ∈ Q, y = x + ax + b}. Then Congruent Numbers

Elliptic Curves P,Q ∈ E(Q) =⇒ P ⊕ Q ∈ E(Q).

The abelian group E(Q) is called the Mordell-Weil group of E/Q.

If the set S = {P, 2P, 3P, ···} is finite, then nP = P for some integer n. Such a point P is called a torsion point, and we denote

the subgroup forme by the torsion points on E(Q) as E(Q)tor.

Anupam Saikia IIT Guwahati July 11, 2020 22 / 28 The Mordell-Weil Group

2 3 Let E(Q) = {(x, y) | x, y ∈ Q, y = x + ax + b}. Then Congruent Numbers

Elliptic Curves P,Q ∈ E(Q) =⇒ P ⊕ Q ∈ E(Q).

The abelian group E(Q) is called the Mordell-Weil group of E/Q.

If the set S = {P, 2P, 3P, ···} is finite, then nP = P for some integer n. Such a point P is called a torsion point, and we denote

the subgroup forme by the torsion points on E(Q) as E(Q)tor.

Anupam Saikia IIT Guwahati July 11, 2020 22 / 28 The Mordell-Weil Group

2 3 Let E(Q) = {(x, y) | x, y ∈ Q, y = x + ax + b}. Then Congruent Numbers

Elliptic Curves P,Q ∈ E(Q) =⇒ P ⊕ Q ∈ E(Q).

The abelian group E(Q) is called the Mordell-Weil group of E/Q.

If the set S = {P, 2P, 3P, ···} is finite, then nP = P for some integer n. Such a point P is called a torsion point, and we denote

the subgroup forme by the torsion points on E(Q) as E(Q)tor.

Anupam Saikia IIT Guwahati July 11, 2020 22 / 28 Mordell-Weil Theorem

Congruent Numbers

Elliptic Curves

Poincare (1901) conjectured that all the points with rational coordinates on an elliptic curve E can obtained by starting with finitely many points and performing addition with them.

Mordell-Weil Theorem: E(Q) is a finitely generated abelian group, i.e.,

rE E(Q) = Z ⊕ E(Q)tor.

The integer rE is called the (algebraic) rank of E over Q.

Anupam Saikia IIT Guwahati July 11, 2020 23 / 28 Mordell-Weil Theorem

Congruent Numbers

Elliptic Curves

Poincare (1901) conjectured that all the points with rational coordinates on an elliptic curve E can obtained by starting with finitely many points and performing addition with them.

Mordell-Weil Theorem: E(Q) is a finitely generated abelian group, i.e.,

rE E(Q) = Z ⊕ E(Q)tor.

The integer rE is called the (algebraic) rank of E over Q.

Anupam Saikia IIT Guwahati July 11, 2020 23 / 28 Mordell-Weil Theorem

Congruent Numbers

Elliptic Curves

Poincare (1901) conjectured that all the points with rational coordinates on an elliptic curve E can obtained by starting with finitely many points and performing addition with them.

Mordell-Weil Theorem: E(Q) is a finitely generated abelian group, i.e.,

rE E(Q) = Z ⊕ E(Q)tor.

The integer rE is called the (algebraic) rank of E over Q.

Anupam Saikia IIT Guwahati July 11, 2020 23 / 28 Mordell-Weil Theorem

Congruent Numbers

Elliptic Curves

Poincare (1901) conjectured that all the points with rational coordinates on an elliptic curve E can obtained by starting with finitely many points and performing addition with them.

Mordell-Weil Theorem: E(Q) is a finitely generated abelian group, i.e.,

rE E(Q) = Z ⊕ E(Q)tor.

The integer rE is called the (algebraic) rank of E over Q.

Anupam Saikia IIT Guwahati July 11, 2020 23 / 28 Torsion and Rank

There are only 15 possibilities for E(Q)tor (Mazur (1978)). Congruent Numbers It is hard to compute the rank of E(Q). Elliptic Curves (Open Question) Can the rank of E(Q) grow arbitrarily large?

We have example of E(Q) with the highest possible rank 28.

The first part of the BSD Conjecture connects E(Q) to an analytic object L(E, s), called the Hasse-Weil L-function of the elliptic curve.

Y −s 1−2s −1 3 L(E, s) ≈ (1 − a p + p ) , s ∈ , Re(s) > . p C 2 p prime

˜ and ap = 1 + p − #E(Fp).

The BSD Conjecture predicts that L(E, 1) = 0 if and only if E(Q) is infinite.

Anupam Saikia IIT Guwahati July 11, 2020 24 / 28 Torsion and Rank

There are only 15 possibilities for E(Q)tor (Mazur (1978)). Congruent Numbers It is hard to compute the rank of E(Q). Elliptic Curves (Open Question) Can the rank of E(Q) grow arbitrarily large?

We have example of E(Q) with the highest possible rank 28.

The first part of the BSD Conjecture connects E(Q) to an analytic object L(E, s), called the Hasse-Weil L-function of the elliptic curve.

Y −s 1−2s −1 3 L(E, s) ≈ (1 − a p + p ) , s ∈ , Re(s) > . p C 2 p prime

˜ and ap = 1 + p − #E(Fp).

The BSD Conjecture predicts that L(E, 1) = 0 if and only if E(Q) is infinite.

Anupam Saikia IIT Guwahati July 11, 2020 24 / 28 Torsion and Rank

There are only 15 possibilities for E(Q)tor (Mazur (1978)). Congruent Numbers It is hard to compute the rank of E(Q). Elliptic Curves (Open Question) Can the rank of E(Q) grow arbitrarily large?

We have example of E(Q) with the highest possible rank 28.

The first part of the BSD Conjecture connects E(Q) to an analytic object L(E, s), called the Hasse-Weil L-function of the elliptic curve.

Y −s 1−2s −1 3 L(E, s) ≈ (1 − a p + p ) , s ∈ , Re(s) > . p C 2 p prime

˜ and ap = 1 + p − #E(Fp).

The BSD Conjecture predicts that L(E, 1) = 0 if and only if E(Q) is infinite.

Anupam Saikia IIT Guwahati July 11, 2020 24 / 28 Torsion and Rank

There are only 15 possibilities for E(Q)tor (Mazur (1978)). Congruent Numbers It is hard to compute the rank of E(Q). Elliptic Curves (Open Question) Can the rank of E(Q) grow arbitrarily large?

We have example of E(Q) with the highest possible rank 28.

The first part of the BSD Conjecture connects E(Q) to an analytic object L(E, s), called the Hasse-Weil L-function of the elliptic curve.

Y −s 1−2s −1 3 L(E, s) ≈ (1 − a p + p ) , s ∈ , Re(s) > . p C 2 p prime

˜ and ap = 1 + p − #E(Fp).

The BSD Conjecture predicts that L(E, 1) = 0 if and only if E(Q) is infinite.

Anupam Saikia IIT Guwahati July 11, 2020 24 / 28 Torsion and Rank

There are only 15 possibilities for E(Q)tor (Mazur (1978)). Congruent Numbers It is hard to compute the rank of E(Q). Elliptic Curves (Open Question) Can the rank of E(Q) grow arbitrarily large?

We have example of E(Q) with the highest possible rank 28.

The first part of the BSD Conjecture connects E(Q) to an analytic object L(E, s), called the Hasse-Weil L-function of the elliptic curve.

Y −s 1−2s −1 3 L(E, s) ≈ (1 − a p + p ) , s ∈ , Re(s) > . p C 2 p prime

˜ and ap = 1 + p − #E(Fp).

The BSD Conjecture predicts that L(E, 1) = 0 if and only if E(Q) is infinite.

Anupam Saikia IIT Guwahati July 11, 2020 24 / 28 Torsion and Rank

There are only 15 possibilities for E(Q)tor (Mazur (1978)). Congruent Numbers It is hard to compute the rank of E(Q). Elliptic Curves (Open Question) Can the rank of E(Q) grow arbitrarily large?

We have example of E(Q) with the highest possible rank 28.

The first part of the BSD Conjecture connects E(Q) to an analytic object L(E, s), called the Hasse-Weil L-function of the elliptic curve.

Y −s 1−2s −1 3 L(E, s) ≈ (1 − a p + p ) , s ∈ , Re(s) > . p C 2 p prime

˜ and ap = 1 + p − #E(Fp).

The BSD Conjecture predicts that L(E, 1) = 0 if and only if E(Q) is infinite.

Anupam Saikia IIT Guwahati July 11, 2020 24 / 28 Torsion and Rank

There are only 15 possibilities for E(Q)tor (Mazur (1978)). Congruent Numbers It is hard to compute the rank of E(Q). Elliptic Curves (Open Question) Can the rank of E(Q) grow arbitrarily large?

We have example of E(Q) with the highest possible rank 28.

The first part of the BSD Conjecture connects E(Q) to an analytic object L(E, s), called the Hasse-Weil L-function of the elliptic curve.

Y −s 1−2s −1 3 L(E, s) ≈ (1 − a p + p ) , s ∈ , Re(s) > . p C 2 p prime

˜ and ap = 1 + p − #E(Fp).

The BSD Conjecture predicts that L(E, 1) = 0 if and only if E(Q) is infinite.

Anupam Saikia IIT Guwahati July 11, 2020 24 / 28 Torsion and Rank

There are only 15 possibilities for E(Q)tor (Mazur (1978)). Congruent Numbers It is hard to compute the rank of E(Q). Elliptic Curves (Open Question) Can the rank of E(Q) grow arbitrarily large?

We have example of E(Q) with the highest possible rank 28.

The first part of the BSD Conjecture connects E(Q) to an analytic object L(E, s), called the Hasse-Weil L-function of the elliptic curve.

Y −s 1−2s −1 3 L(E, s) ≈ (1 − a p + p ) , s ∈ , Re(s) > . p C 2 p prime

˜ and ap = 1 + p − #E(Fp).

The BSD Conjecture predicts that L(E, 1) = 0 if and only if E(Q) is infinite.

Anupam Saikia IIT Guwahati July 11, 2020 24 / 28 Progress in the Congruent Number Problem

A rational point (x, y) on the congruent number elliptic curve Congruent 2 3 2 Numbers En : y = x − n x has infinite order if and only if y 6= 0.

Elliptic Curves Thus, n is a congruent number if and only if En(Q) is infinite.

A result of Nagell (1929) shows that Ep(Q) is finite for any prime p of the form 8k + 3, so such primes are non-congruent.

For any integer n of the form 8k + 5, 8k + 6 and 8k + 7, L(En, 1) = 0, hence En(Q) should be infinite by the BSD Conjecture and n should be congruent.

Heegner (1952) showed how to construct a point of infinite order on E2p(Q) for any prime p of the form 8k + 3. Monsky (1990) extended this method construct point of infinite order on Ep for a prime p of the form 8k + 5 and 8k + 7, thereby proving that such primes are congruent. Tian (2012) proved it for highly composite numbers.

Anupam Saikia IIT Guwahati July 11, 2020 25 / 28 Progress in the Congruent Number Problem

A rational point (x, y) on the congruent number elliptic curve Congruent 2 3 2 Numbers En : y = x − n x has infinite order if and only if y 6= 0.

Elliptic Curves Thus, n is a congruent number if and only if En(Q) is infinite.

A result of Nagell (1929) shows that Ep(Q) is finite for any prime p of the form 8k + 3, so such primes are non-congruent.

For any integer n of the form 8k + 5, 8k + 6 and 8k + 7, L(En, 1) = 0, hence En(Q) should be infinite by the BSD Conjecture and n should be congruent.

Heegner (1952) showed how to construct a point of infinite order on E2p(Q) for any prime p of the form 8k + 3. Monsky (1990) extended this method construct point of infinite order on Ep for a prime p of the form 8k + 5 and 8k + 7, thereby proving that such primes are congruent. Tian (2012) proved it for highly composite numbers.

Anupam Saikia IIT Guwahati July 11, 2020 25 / 28 Progress in the Congruent Number Problem

A rational point (x, y) on the congruent number elliptic curve Congruent 2 3 2 Numbers En : y = x − n x has infinite order if and only if y 6= 0.

Elliptic Curves Thus, n is a congruent number if and only if En(Q) is infinite.

A result of Nagell (1929) shows that Ep(Q) is finite for any prime p of the form 8k + 3, so such primes are non-congruent.

For any integer n of the form 8k + 5, 8k + 6 and 8k + 7, L(En, 1) = 0, hence En(Q) should be infinite by the BSD Conjecture and n should be congruent.

Heegner (1952) showed how to construct a point of infinite order on E2p(Q) for any prime p of the form 8k + 3. Monsky (1990) extended this method construct point of infinite order on Ep for a prime p of the form 8k + 5 and 8k + 7, thereby proving that such primes are congruent. Tian (2012) proved it for highly composite numbers.

Anupam Saikia IIT Guwahati July 11, 2020 25 / 28 Progress in the Congruent Number Problem

A rational point (x, y) on the congruent number elliptic curve Congruent 2 3 2 Numbers En : y = x − n x has infinite order if and only if y 6= 0.

Elliptic Curves Thus, n is a congruent number if and only if En(Q) is infinite.

A result of Nagell (1929) shows that Ep(Q) is finite for any prime p of the form 8k + 3, so such primes are non-congruent.

For any integer n of the form 8k + 5, 8k + 6 and 8k + 7, L(En, 1) = 0, hence En(Q) should be infinite by the BSD Conjecture and n should be congruent.

Heegner (1952) showed how to construct a point of infinite order on E2p(Q) for any prime p of the form 8k + 3. Monsky (1990) extended this method construct point of infinite order on Ep for a prime p of the form 8k + 5 and 8k + 7, thereby proving that such primes are congruent. Tian (2012) proved it for highly composite numbers.

Anupam Saikia IIT Guwahati July 11, 2020 25 / 28 Progress in the Congruent Number Problem

A rational point (x, y) on the congruent number elliptic curve Congruent 2 3 2 Numbers En : y = x − n x has infinite order if and only if y 6= 0.

Elliptic Curves Thus, n is a congruent number if and only if En(Q) is infinite.

A result of Nagell (1929) shows that Ep(Q) is finite for any prime p of the form 8k + 3, so such primes are non-congruent.

For any integer n of the form 8k + 5, 8k + 6 and 8k + 7, L(En, 1) = 0, hence En(Q) should be infinite by the BSD Conjecture and n should be congruent.

Heegner (1952) showed how to construct a point of infinite order on E2p(Q) for any prime p of the form 8k + 3. Monsky (1990) extended this method construct point of infinite order on Ep for a prime p of the form 8k + 5 and 8k + 7, thereby proving that such primes are congruent. Tian (2012) proved it for highly composite numbers.

Anupam Saikia IIT Guwahati July 11, 2020 25 / 28 Progress in the Congruent Number Problem

A rational point (x, y) on the congruent number elliptic curve Congruent 2 3 2 Numbers En : y = x − n x has infinite order if and only if y 6= 0.

Elliptic Curves Thus, n is a congruent number if and only if En(Q) is infinite.

A result of Nagell (1929) shows that Ep(Q) is finite for any prime p of the form 8k + 3, so such primes are non-congruent.

For any integer n of the form 8k + 5, 8k + 6 and 8k + 7, L(En, 1) = 0, hence En(Q) should be infinite by the BSD Conjecture and n should be congruent.

Heegner (1952) showed how to construct a point of infinite order on E2p(Q) for any prime p of the form 8k + 3. Monsky (1990) extended this method construct point of infinite order on Ep for a prime p of the form 8k + 5 and 8k + 7, thereby proving that such primes are congruent. Tian (2012) proved it for highly composite numbers.

Anupam Saikia IIT Guwahati July 11, 2020 25 / 28 Progress in the Congruent Number Problem

A rational point (x, y) on the congruent number elliptic curve Congruent 2 3 2 Numbers En : y = x − n x has infinite order if and only if y 6= 0.

Elliptic Curves Thus, n is a congruent number if and only if En(Q) is infinite.

A result of Nagell (1929) shows that Ep(Q) is finite for any prime p of the form 8k + 3, so such primes are non-congruent.

For any integer n of the form 8k + 5, 8k + 6 and 8k + 7, L(En, 1) = 0, hence En(Q) should be infinite by the BSD Conjecture and n should be congruent.

Heegner (1952) showed how to construct a point of infinite order on E2p(Q) for any prime p of the form 8k + 3. Monsky (1990) extended this method construct point of infinite order on Ep for a prime p of the form 8k + 5 and 8k + 7, thereby proving that such primes are congruent. Tian (2012) proved it for highly composite numbers.

Anupam Saikia IIT Guwahati July 11, 2020 25 / 28 A Slide on My Research in Related Topics

Elliptic Curves Congruent Numbers

Elliptic Curves – Torsion subgroup of elliptic curves over number fields. – Selmer groups - they contain arithmetic information about the elliptic curve including the Mordell-Weil group.

Congruent Numbers

– Construction of infinite families of highly composite non-congruent numbers. – θ-congruent numbers, a generalization of congruent numbers for an angle with rational cosine rather than just π/2.

Anupam Saikia IIT Guwahati July 11, 2020 26 / 28 A Slide on My Research in Related Topics

Elliptic Curves Congruent Numbers

Elliptic Curves – Torsion subgroup of elliptic curves over number fields. – Selmer groups - they contain arithmetic information about the elliptic curve including the Mordell-Weil group.

Congruent Numbers

– Construction of infinite families of highly composite non-congruent numbers. – θ-congruent numbers, a generalization of congruent numbers for an angle with rational cosine rather than just π/2.

Anupam Saikia IIT Guwahati July 11, 2020 26 / 28 A Slide on My Research in Related Topics

Elliptic Curves Congruent Numbers

Elliptic Curves – Torsion subgroup of elliptic curves over number fields. – Selmer groups - they contain arithmetic information about the elliptic curve including the Mordell-Weil group.

Congruent Numbers

– Construction of infinite families of highly composite non-congruent numbers. – θ-congruent numbers, a generalization of congruent numbers for an angle with rational cosine rather than just π/2.

Anupam Saikia IIT Guwahati July 11, 2020 26 / 28 A Slide on My Research in Related Topics

Elliptic Curves Congruent Numbers

Elliptic Curves – Torsion subgroup of elliptic curves over number fields. – Selmer groups - they contain arithmetic information about the elliptic curve including the Mordell-Weil group.

Congruent Numbers

– Construction of infinite families of highly composite non-congruent numbers. – θ-congruent numbers, a generalization of congruent numbers for an angle with rational cosine rather than just π/2.

Anupam Saikia IIT Guwahati July 11, 2020 26 / 28 References

B. Birch and P. Swinnerton-Dyer, Notes on elliptic curves II, Crelle Congruent 218 (1965) 79–108. Numbers Elliptic Curves J. Coates, Congruent Numbers, PNAS 109 (2012), 21182–21183.

N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer 1993.

N. Koblitz, A Course in Number Theory and Cryptography, Springer 1994.

P. Monsky, Mock Heegner points and congruent numbers, Math. Z. 204 (1990), no. 1, 45–67.

Y. Tian Congruent numbers with many prime factors, PNAS 109 (2012), 211256–21258.

Anupam Saikia IIT Guwahati July 11, 2020 27 / 28 Congruent Numbers

Elliptic Curves

THANK YOU

Anupam Saikia IIT Guwahati July 11, 2020 28 / 28