MATH36032 Problem Solving by Computer Exploring numbers I There are infinitely many prime numbers I (Fermat's Last Theorem:) There is not no non-trivial integer solutions to x n + y n = zn for n ≥ 3 I (Fermat's Little Theorem:) if p is a prime, then for any number a co-prime to p, then ap−1 ≡ 1 (mod p). I Famous (proved) theorems in number theory I Special numbers: Mersenne prime 2p − 1, Fermat number 2n Fn = 2 + 1, ··· I Important applications: Random number generation, cryptography (RSA public-key algorithm) Integers and prime numbers I Many problems are simple to state, yet difficult to prove or refute I Special numbers: Mersenne prime 2p − 1, Fermat number 2n Fn = 2 + 1, ··· I Important applications: Random number generation, cryptography (RSA public-key algorithm) Integers and prime numbers I Many problems are simple to state, yet difficult to prove or refute I Famous (proved) theorems in number theory I There are infinitely many prime numbers I (Fermat's Last Theorem:) There is not no non-trivial integer solutions to x n + y n = zn for n ≥ 3 I (Fermat's Little Theorem:) if p is a prime, then for any number a co-prime to p, then ap−1 ≡ 1 (mod p). I Important applications: Random number generation, cryptography (RSA public-key algorithm) Integers and prime numbers I Many problems are simple to state, yet difficult to prove or refute I Famous (proved) theorems in number theory I There are infinitely many prime numbers I (Fermat's Last Theorem:) There is not no non-trivial integer solutions to x n + y n = zn for n ≥ 3 I (Fermat's Little Theorem:) if p is a prime, then for any number a co-prime to p, then ap−1 ≡ 1 (mod p). I Special numbers: Mersenne prime 2p − 1, Fermat number 2n Fn = 2 + 1, ··· Integers and prime numbers I Many problems are simple to state, yet difficult to prove or refute I Famous (proved) theorems in number theory I There are infinitely many prime numbers I (Fermat's Last Theorem:) There is not no non-trivial integer solutions to x n + y n = zn for n ≥ 3 I (Fermat's Little Theorem:) if p is a prime, then for any number a co-prime to p, then ap−1 ≡ 1 (mod p). I Special numbers: Mersenne prime 2p − 1, Fermat number 2n Fn = 2 + 1, ··· I Important applications: Random number generation, cryptography (RSA public-key algorithm) Unproved conjectures in number theory I Goldbach Conjecture: every even integer greater than two is the sum of two prime numbers (tested by for all integer up to 1020) I Twin Prime Conjecture: are there infinitely many pairs of twin primes like (5; 7); (29; 31); ··· ? Largest twin prime pair known 2996863034895 · 21290000 ± 1 (388; 342 digits) I Collatz Conjectures: For any integer n, the iteration ( 3n + 1; if n is odd; n 7! n=2; if n is even; eventually contains 1. I Riemann Hypothesis: All non-trivial zeroes of the zeta function 1 X ζ(s) = n−s n=1 have real part 1=2 (on the complex plane). Two "simple" questions in the introduction Q1: what is the smallest (in absolute value) integer solutions to x3 + y 3 + z3 = 42? x = −80538738812075974; y = 80435758145817515; z = 12602123297335631: Q2: what is the smallest non-negative integer solutions to x y z + + = 4? y + z z + x x + y x = 154476802108746166441951315019919837485664325669565431700026634898253202035277999; y = 36875131794129999827197811565225474825492979968971970996283137471637224634055579; z = 4373612677928697257861252602371390152816537558161613618621437993378423467772036: Some related built-in functions I primes(N): return all primes up to N I isprime: whether a number is prime I factor: factorize the number I gcd: greatest common divisor I lcm: least common multiplier Each type has its own range, for example: I int8: [−27; 27 − 1] = [−128; 127] I int64: [−263; 263 − 1] = [−9223372036854775808; 9223372036854775807] (19 digits) I uint16: [0; 216 − 1] = [0; 65535] I double: about [−253; 253](≈ 9 × 1015) I Many operations (like sqrt) can not be applied to integers. I Make sure the number stays in the right range (otherwise the number is shrinked to the range) I Use symbolic numbers (though slower), if outside this range, like the factorization of 20!: factor(factorial(sym(20))) Representing numbers in MATLAB I double: the default type for any decimal number, like the output for rand I int8, int16, int32, int64: signed integers I uint8, uint16, uint32, uint64: unsigned integers I Many operations (like sqrt) can not be applied to integers. I Make sure the number stays in the right range (otherwise the number is shrinked to the range) I Use symbolic numbers (though slower), if outside this range, like the factorization of 20!: factor(factorial(sym(20))) Representing numbers in MATLAB I double: the default type for any decimal number, like the output for rand I int8, int16, int32, int64: signed integers I uint8, uint16, uint32, uint64: unsigned integers Each type has its own range, for example: I int8: [−27; 27 − 1] = [−128; 127] I int64: [−263; 263 − 1] = [−9223372036854775808; 9223372036854775807] (19 digits) I uint16: [0; 216 − 1] = [0; 65535] I double: about [−253; 253](≈ 9 × 1015) Representing numbers in MATLAB I double: the default type for any decimal number, like the output for rand I int8, int16, int32, int64: signed integers I uint8, uint16, uint32, uint64: unsigned integers Each type has its own range, for example: I int8: [−27; 27 − 1] = [−128; 127] I int64: [−263; 263 − 1] = [−9223372036854775808; 9223372036854775807] (19 digits) I uint16: [0; 216 − 1] = [0; 65535] I double: about [−253; 253](≈ 9 × 1015) I Many operations (like sqrt) can not be applied to integers. I Make sure the number stays in the right range (otherwise the number is shrinked to the range) I Use symbolic numbers (though slower), if outside this range, like the factorization of 20!: factor(factorial(sym(20))) Prime number theorem Problem: verify that the number of prime up N is about N= log N, when N is large? How do you use the built-in function primes to accomplish this? Related Problem: find the largest twin primes both less than or equal to N Twin prime conjecture: are there infinitely many twin primes? Twin primes Problem: List all twin primes up to N Find all pairs of consecutive primes in the output of primes, with gap exactly two. Twin prime conjecture: are there infinitely many twin primes? Twin primes Problem: List all twin primes up to N Find all pairs of consecutive primes in the output of primes, with gap exactly two. Related Problem: find the largest twin primes both less than or equal to N Twin primes Problem: List all twin primes up to N Find all pairs of consecutive primes in the output of primes, with gap exactly two. Related Problem: find the largest twin primes both less than or equal to N Twin prime conjecture: are there infinitely many twin primes? Largest gap between consecutive primes 1 Problems: what is the largest gap between two consecutive primes that are less than or equal to a given number N 1More on wiki: https://en.wikipedia.org/wiki/Prime_gap Unfortunate, Fn with n ≤ 4 are the only known prime numbers; the conjecture was refuted first by Euler in 1732 by showing that 32 F5 = 2 + 1 = 4294967297 = 641 × 6700417: Current Fn is completely factorized up to n = 11, and incompletely factorized for a few more (only one or more factors of Fn are known). 2n Fermat prime number: Fn = 2 + 1 Observations made by Pierre de Fermat: 1 F0 = 2 + 1 =3 is prime 2 F1 = 2 + 1 =5 is prime 22 F2 = 2 + 1=17 is prime 23 F3 = 2 + 1=257 is prime 24 F4 = 2 + 1=65537 is prime He concluded all such numbers are prime. Current Fn is completely factorized up to n = 11, and incompletely factorized for a few more (only one or more factors of Fn are known). 2n Fermat prime number: Fn = 2 + 1 Observations made by Pierre de Fermat: 1 F0 = 2 + 1 =3 is prime 2 F1 = 2 + 1 =5 is prime 22 F2 = 2 + 1=17 is prime 23 F3 = 2 + 1=257 is prime 24 F4 = 2 + 1=65537 is prime He concluded all such numbers are prime. Unfortunate, Fn with n ≤ 4 are the only known prime numbers; the conjecture was refuted first by Euler in 1732 by showing that 32 F5 = 2 + 1 = 4294967297 = 641 × 6700417: 2n Fermat prime number: Fn = 2 + 1 Observations made by Pierre de Fermat: 1 F0 = 2 + 1 =3 is prime 2 F1 = 2 + 1 =5 is prime 22 F2 = 2 + 1=17 is prime 23 F3 = 2 + 1=257 is prime 24 F4 = 2 + 1=65537 is prime He concluded all such numbers are prime. Unfortunate, Fn with n ≤ 4 are the only known prime numbers; the conjecture was refuted first by Euler in 1732 by showing that 32 F5 = 2 + 1 = 4294967297 = 641 × 6700417: Current Fn is completely factorized up to n = 11, and incompletely factorized for a few more (only one or more factors of Fn are known). Make sure the right operation is applied on the string or vector (convert back into numbers), like the built-in function str2num. Numbers and Digits num2str: conversion of a number like 2020 into string (sequence of characters). You can convert digital string back to numbers, like 1 str2num('2020'); % the output is the number 2020 Equivalently, how to convert a number into a vector of number (from the number 2020 to [2, 0, 2, 0])? 1 function dig = num2dig(n) 2 % convert a number into a vector of digits 3 % for example: num2dig(2020) --> [2 0 2 0] Numbers and Digits num2str: conversion of a number like 2020 into string (sequence of characters).