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Geometric Approaches to Solving Diophantine Equations

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Geometric Approaches to Solving Diophantine Equations Geometric approaches to solving Diophantine equations Alex J. Best Introduction Geometric approaches to solving Geometry of Numbers Diophantine equations Rational points on surfaces Alex J. Best Tomorrows Mathematicians Today 2013 16-2-2013 I x2 + y 2 = z2 (Pythagorean triples) I x2 − ny 2 = 1 (Pell's equation) I x3 + 48 = y 4 I 9x − 8y = 1 Geometric Introduction approaches to solving Diophantine equations Named for Diophantus of Alexandria (≈ 250AD) Alex J. Best Some Diophantine equations: Introduction Geometry of Numbers Rational points on surfaces I x2 − ny 2 = 1 (Pell's equation) I x3 + 48 = y 4 I 9x − 8y = 1 Geometric Introduction approaches to solving Diophantine equations Named for Diophantus of Alexandria (≈ 250AD) Alex J. Best Some Diophantine equations: Introduction I 2 2 2 Geometry of x + y = z Numbers Rational points on (Pythagorean triples) surfaces I x3 + 48 = y 4 I 9x − 8y = 1 Geometric Introduction approaches to solving Diophantine equations Named for Diophantus of Alexandria (≈ 250AD) Alex J. Best Some Diophantine equations: Introduction I 2 2 2 Geometry of x + y = z Numbers Rational points on (Pythagorean triples) surfaces I x2 − ny 2 = 1 (Pell's equation) I 9x − 8y = 1 Geometric Introduction approaches to solving Diophantine equations Named for Diophantus of Alexandria (≈ 250AD) Alex J. Best Some Diophantine equations: Introduction I 2 2 2 Geometry of x + y = z Numbers Rational points on (Pythagorean triples) surfaces I x2 − ny 2 = 1 (Pell's equation) I x3 + 48 = y 4 Geometric Introduction approaches to solving Diophantine equations Named for Diophantus of Alexandria (≈ 250AD) Alex J. Best Some Diophantine equations: Introduction I 2 2 2 Geometry of x + y = z Numbers Rational points on (Pythagorean triples) surfaces I x2 − ny 2 = 1 (Pell's equation) I x3 + 48 = y 4 I 9x − 8y = 1 I Are there any solutions? I How many are there? I Can we classify them? I How can we compute them? Geometric What do we want to know? approaches to solving Diophantine equations Alex J. Best Introduction Geometry of Numbers Rational points on surfaces I How many are there? I Can we classify them? I How can we compute them? Geometric What do we want to know? approaches to solving Diophantine equations Alex J. Best Introduction Geometry of Numbers Rational points on I Are there any solutions? surfaces I Can we classify them? I How can we compute them? Geometric What do we want to know? approaches to solving Diophantine equations Alex J. Best Introduction Geometry of Numbers Rational points on I Are there any solutions? surfaces I How many are there? I How can we compute them? Geometric What do we want to know? approaches to solving Diophantine equations Alex J. Best Introduction Geometry of Numbers Rational points on I Are there any solutions? surfaces I How many are there? I Can we classify them? Geometric What do we want to know? approaches to solving Diophantine equations Alex J. Best Introduction Geometry of Numbers Rational points on I Are there any solutions? surfaces I How many are there? I Can we classify them? I How can we compute them? I Geometry of numbers I Rational points on surfaces Geometric In this talk approaches to solving Diophantine equations Alex J. Best Introduction Geometry of Numbers Rational points on Two general ideas: surfaces I Rational points on surfaces Geometric In this talk approaches to solving Diophantine equations Alex J. Best Introduction Geometry of Numbers Rational points on Two general ideas: surfaces I Geometry of numbers Geometric In this talk approaches to solving Diophantine equations Alex J. Best Introduction Geometry of Numbers Rational points on Two general ideas: surfaces I Geometry of numbers I Rational points on surfaces Geometric Minkowski and the Geometry of Numbers approaches to solving Diophantine equations 1910 - Hermann Minkowski publishes his paper \Geometrie Alex J. Best der Zahlen" and sparks a new field called the Geometry of Numbers. Introduction Geometry of Numbers Rational points on surfaces We can try a few primes, and notice that 22 + 12 = 5, 32 + 22 = 13, 42 + 12 = 17, ... all work. But 7, 11, 19, ... do not. So maybe x2 + y 2 = p when p ≡ 1 (mod 4). Theorem (Fermat's Christmas Theorem) An odd prime p can be written as p = x2 + y 2 if and only if p ≡ 1 (mod 4). Geometric A motivating example approaches to solving Diophantine equations Alex J. Best Introduction We want to find when integers x; y such that x2 + y 2 = p Geometry of where p is prime. Numbers Rational points on surfaces But 7, 11, 19, ... do not. So maybe x2 + y 2 = p when p ≡ 1 (mod 4). Theorem (Fermat's Christmas Theorem) An odd prime p can be written as p = x2 + y 2 if and only if p ≡ 1 (mod 4). Geometric A motivating example approaches to solving Diophantine equations Alex J. Best Introduction We want to find when integers x; y such that x2 + y 2 = p Geometry of where p is prime. Numbers We can try a few primes, and notice that 22 + 12 = 5, Rational points on surfaces 32 + 22 = 13, 42 + 12 = 17, ... all work. So maybe x2 + y 2 = p when p ≡ 1 (mod 4). Theorem (Fermat's Christmas Theorem) An odd prime p can be written as p = x2 + y 2 if and only if p ≡ 1 (mod 4). Geometric A motivating example approaches to solving Diophantine equations Alex J. Best Introduction We want to find when integers x; y such that x2 + y 2 = p Geometry of where p is prime. Numbers We can try a few primes, and notice that 22 + 12 = 5, Rational points on surfaces 32 + 22 = 13, 42 + 12 = 17, ... all work. But 7, 11, 19, ... do not. Geometric A motivating example approaches to solving Diophantine equations Alex J. Best Introduction We want to find when integers x; y such that x2 + y 2 = p Geometry of where p is prime. Numbers We can try a few primes, and notice that 22 + 12 = 5, Rational points on surfaces 32 + 22 = 13, 42 + 12 = 17, ... all work. But 7, 11, 19, ... do not. So maybe x2 + y 2 = p when p ≡ 1 (mod 4). Theorem (Fermat's Christmas Theorem) An odd prime p can be written as p = x2 + y 2 if and only if p ≡ 1 (mod 4). Geometric What can geometry do for us? approaches to solving Diophantine When p = 5 we can look at points satisfying our criteria: equations Alex J. Best Introduction Geometry of Numbers Rational points on surfaces Geometric What can geometry do for us? approaches to solving Diophantine When p = 5 we can look at points satisfying our criteria: equations Alex J. Best Introduction Geometry of Numbers Rational points on surfaces Geometric What can geometry do for us? approaches to solving Diophantine When p = 5 we can look at points satisfying our criteria: equations Alex J. Best Introduction Geometry of Numbers Rational points on surfaces Geometric What can geometry do for us? approaches to solving Diophantine When p = 5 we can look at points satisfying our criteria: equations Alex J. Best Introduction Geometry of Numbers Rational points on surfaces Theorem (Minkowski) n If B is a convex symmetric body and Λ a lattice in R then B contains a non-zero point of the lattice if: Vol(B) > 2d i Where i is the area of a single cell of the lattice Λ. We can find it using determinants, or the order of our lattice as a n subgroup of Z for the more algebraically inclined. Geometric Minkowski's first theorem approaches to solving Diophantine equations Alex J. Best n We say a set B in R is symmetric if x 2 B =) −x 2 B. Introduction We say a set B is n is convex if R Geometry of x; y 2 B =) x + λ(y − x) 2 B for 0 ≤ λ ≤ 1. Numbers Rational points on surfaces Geometric Minkowski's first theorem approaches to solving Diophantine equations Alex J. Best n We say a set B in R is symmetric if x 2 B =) −x 2 B. Introduction We say a set B is n is convex if R Geometry of x; y 2 B =) x + λ(y − x) 2 B for 0 ≤ λ ≤ 1. Numbers Rational points on Theorem (Minkowski) surfaces n If B is a convex symmetric body and Λ a lattice in R then B contains a non-zero point of the lattice if: Vol(B) > 2d i Where i is the area of a single cell of the lattice Λ. We can find it using determinants, or the order of our lattice as a n subgroup of Z for the more algebraically inclined. The area of our lattice cell is p. The area of our circle is π2p So Minkowski tells us that as: π2p = Vol(B) > 2d i = 22p = 4p We have a point which satisfies x2 + y 2 = kp for some k ≥ 1, and also x2 + y 2 < 2p. So x2 + y 2 = p and we are done. Geometric Let's apply it approaches to solving Diophantine equations Alex J. Best Introduction 2 A circle in R is symmetric and convex, great! Geometry of Numbers Rational points on surfaces The area of our circle is π2p So Minkowski tells us that as: π2p = Vol(B) > 2d i = 22p = 4p We have a point which satisfies x2 + y 2 = kp for some k ≥ 1, and also x2 + y 2 < 2p. So x2 + y 2 = p and we are done. Geometric Let's apply it approaches to solving Diophantine equations Alex J. Best Introduction 2 A circle in R is symmetric and convex, great! Geometry of The area of our lattice cell is p.
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