intro_number_theory

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Introduction to Number Theory 1

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Preview Number Theory Essentials Congruence classes, Modular arithmetic Prime numbers challenges Fermat’s Little theorem he Totient function The Totient function Euler's Theorem Quadratic residuocity Foundation of RSA 2
Number Theory Essentials Prime Numbers A number a I is a prime iff p it's only factors are itself and 1 Equivalently , x I , gcd ( x,a ) = 1 a, b I are relatively prime iff : gcd ( a,b ) = 1 • Fundamental theorem of arithmetic: Every integer has a unique factorization that is a 3 product of prime powers.

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C l Congruence Classes: e integers modulo 5 the integers modulo 5 . . . 5 10 6 0 4 1 9 11 14 ……. . 1 6 11 (mod 5) ……. . 8 7 3 2 4 12 13 ……. . ……. .
Modular arithmetic Form: a b mod n he modulo relation partitions the integers The modulo relation partitions the integers into congruence classes The congruence class of an integer ' a ' is the set of all integers congruent to ' a ' modulo ' n '. a b mod n asserts that ' a ' and ' b ' are members of the same congruence class modulo ' n ' 5

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The integers modulo n a,b,n I, a b mod n iff n | ( a-b ) * 8 mod 11: (28 )/11 = 2 28 6 mod 11: (28-6)/11 = 2 I 219 49 mod 17: (219-49)/17 = 10 I Symmetry : If a b mod n then b a mod n Transitivity : od d od en od If a b mod n and b c mod n then a c mod n 6
Modular arithmetic: otation notation Form: a

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