Math 412: Number Theory Lecture 3: Prime Decomposition of
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1 Math 412: Number Theory Lecture 3: Prime Decomposition of Integers Gexin Yu College of William and Mary
2 Prime numbers Definition: a (positive) integer p is prime if p has no divisor other than 1 and p. Otherwise it is a composite number. Ex: Every composite number has a prime divisor. Let p be a prime. If p ab, then p a or p b. Ex: every composite integer n has a prime divisor at most n.
3 Ex: there are infinite many prime numbers.
4 Ex: there are infinite many prime numbers. Ex: there are infinite many primes of form 4k + 3.
5 Two special kinds of primes:
6 Two special kinds of primes: a prime of the form 2 p 1 is called a Mersenne prime;
7 Two special kinds of primes: a prime of the form 2 p 1 is called a Mersenne prime; a prime of the form 2 2n + 1 is the called a Fermat Prime.
8 Two special kinds of primes: a prime of the form 2 p 1 is called a Mersenne prime; a prime of the form 2 2n + 1 is the called a Fermat Prime. Great Internet Mersenne Prime Search (GIMPS): search for larger Mersenne prime. Currently only 49 are known and the largest is M with 22,338,618 digits (found in Jan. 07, 2016).
9 Two special kinds of primes: a prime of the form 2 p 1 is called a Mersenne prime; a prime of the form 2 2n + 1 is the called a Fermat Prime. Great Internet Mersenne Prime Search (GIMPS): search for larger Mersenne prime. Currently only 49 are known and the largest is M with 22,338,618 digits (found in Jan. 07, 2016). Famous Conejectures on Primes:
10 Two special kinds of primes: a prime of the form 2 p 1 is called a Mersenne prime; a prime of the form 2 2n + 1 is the called a Fermat Prime. Great Internet Mersenne Prime Search (GIMPS): search for larger Mersenne prime. Currently only 49 are known and the largest is M with 22,338,618 digits (found in Jan. 07, 2016). Famous Conejectures on Primes: Mersenne primes: there are infinitely many Mersenne primes;
11 Two special kinds of primes: a prime of the form 2 p 1 is called a Mersenne prime; a prime of the form 2 2n + 1 is the called a Fermat Prime. Great Internet Mersenne Prime Search (GIMPS): search for larger Mersenne prime. Currently only 49 are known and the largest is M with 22,338,618 digits (found in Jan. 07, 2016). Famous Conejectures on Primes: Mersenne primes: there are infinitely many Mersenne primes; Fermat primes: there are only finitely many Fermat primes;
12 Two special kinds of primes: a prime of the form 2 p 1 is called a Mersenne prime; a prime of the form 2 2n + 1 is the called a Fermat Prime. Great Internet Mersenne Prime Search (GIMPS): search for larger Mersenne prime. Currently only 49 are known and the largest is M with 22,338,618 digits (found in Jan. 07, 2016). Famous Conejectures on Primes: Mersenne primes: there are infinitely many Mersenne primes; Fermat primes: there are only finitely many Fermat primes; Twin prime conjecture: there are infinitely many primes p such that p + 2 is also prime;
13 Two special kinds of primes: a prime of the form 2 p 1 is called a Mersenne prime; a prime of the form 2 2n + 1 is the called a Fermat Prime. Great Internet Mersenne Prime Search (GIMPS): search for larger Mersenne prime. Currently only 49 are known and the largest is M with 22,338,618 digits (found in Jan. 07, 2016). Famous Conejectures on Primes: Mersenne primes: there are infinitely many Mersenne primes; Fermat primes: there are only finitely many Fermat primes; Twin prime conjecture: there are infinitely many primes p such that p + 2 is also prime; Goldbach Conjecture: every even number can be written as a sum of two primes.
14 Fundamental Theorem of Arithmetic FTA: every integer can be uniquely written as a product of primes.
15 Ex: let [a, b] be the least common multiplier of a and b. show that [a, b](a, b) = ab.
16 Ex: If (m, n) = 1 and d mn. Then there exist unique pair d 1, d 2 so that d 1 m and d 2 n and d = d 1 d 2. Conversely, if d 1 m and d 2 n, then d 1 d 2 mn.
17 Ex: if (a, b) = 1 and ab = n 2 for some n Z, then a and b are squares.
18 Ex: Let α be a real number that is a root of the polynomial x n + c n 1 x n c 1 x + c 0, where c i Z. Then α is either an integer or an irrational number. (such an α is algebraic, but not transcendental)
19 Ex: If N is a nonsquare natural number, then N is irrational.
20 Ex: If N is a nonsquare natural number, then N is irrational. Ex: show that is an irrational number.
21 Ex: show that log 3 5 is irrational.
22 Ex: if s is a real number with s > 1, then θ(s) = n=1 1 n s = p prime (1 1 p s ) 1.
23 Ex: Show that p ( p k) for any prime p and 1 k < p.
24 Ex: If a k b and a k+1 b, then we write a k b. Let α = α(p, n!) be p α n!. Then α = j=1 n p j and n! = p α(p,n!) p n
25 Ex: If a k b and a k+1 b, then we write a k b. Let α = α(p, n!) be p α n!. Then α = j=1 n p j and n! = p α(p,n!) p n Ex: How many zeros are in the end of 20!? Write the prime factorization of 20!.
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