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Summary of Number Theory

Summary of Number Theory. Basics. Fundamental Theorem of Arithmetic Every positive integer > 1 can be written as a product of primes. A corollary says that these can be written uniquely in a canonical form Euclid’s Theorem There an infinite number of primes Number of Primes in a range

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Summary of Number Theory

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  1. Summary of Number Theory

  2. Basics • Fundamental Theorem of Arithmetic • Every positive integer > 1 can be written as a product of primes. A corollary says that these can be written uniquely in a canonical form • Euclid’s Theorem • There an infinite number of primes • Number of Primes in a range • Relative primality and GCD • Division algorithm

  3. More Primes • Let a,b be two integers with at least one of a,b non-zero and d = gcd(a,b). Then there exist integers, x, y such that ax+by = d and if a and b are relatively prime, ax+by = 1 • Euclid’s lemma: if a|bc with gcd(a,b) = 1, then a|ac • If p is prime and p|ab, then p|a or p|b

  4. Congruence • Definition • 4 properties of congruence (e.g., a is congruent 0 mod n iff n|a) • Addition, subtraction, multiplication, transitivity of congruences • Conditions under which divisions holds • If x, y, n p are integers and x is congruent to y mod n, then x is congruent to y + pn mod n

  5. Multiplicative Inverses • Def. • Suppose gcd(a,n) = 1. Then there exists integers, s, t such that as + nt = 1. S is the multiplicative inverse of a mod n • Finding the multiplicative inverse

  6. Important Theorems • Fermat’s little theorem (if p is prime and p does not divide a, ap-1 is congruent to 1 mod p) • Using Fermat to test primality • Euler’s Phi function • If p is prime and k > 0, phi(pk) = pk(1 – 1/p) • Chinese Remainder theorem • Euler’s Theorem (if gcd(a,n) = 1, aphi(n) is congruent to 1 mod n)

  7. More • Def. of a number theoretic function • Computing phi(n) from n’s prime factorization • Order • Let n > 1 and gcd(a,n) = 1. The order of a mod n is the smallest positive integer, k, such that ak is congruent to 1 mod n • Primitive Root • If gcd(a,n) – 1 and a is the of order phi(n) mod n, then a is a primitive root of n • The discrete Log problem • A composite number, a > 1, will always possess a prime divisor, p, <= square root of a • Let n be an integer and suppose there exist integers x and y with x2 is congruent to y2 but x is not congruent to +/- y mod n, then n is composite and gcd (x-y,n) gives a non-trivial factor of n

  8. Yet More • Def. of a pseudo-prime: some n that satisfies the converse of Fermat’s little theorem • Let integer a have k mod n. The ah is congruent to 1 mod n iff k|h. • If n has a primitieve root, then it has exactly phi(phi(n)) of them • If a has order k mod p, then ai is congruent to aj mod p iff i is congruent to j mod k

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