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Prime numbers are widely studied in the field of number theory. One approach to investigate prime numbers is to study numbers of a certain form. For example, it has been proven that there are infinitely many primes in the form a + nd, where d ≥ 2 and gcd(d, a) = 1 (Dirichlet’s theorem).
Here are some properties of the Fermat numbers. Proposition. If p is prime and p | Fn, then p = k · 2n+2 + 1 for some k. I won’t prove this result, since the proof requires results about quadratic residues which I won’t discuss for a while. Here’s how it can be used. Example. Check F4 = 224+ 1 = 65537 for primality.
De nition 4.1. The number F(n) = 22n + 1 is called a Fermat number; and if it is prime it is called a Fermat prime. 19
Fermat's little theorem is probably the most useful and frequently used tool in number theory. Fermat claimed that any prime number p has to satisfy aP - 1 =" 1 (mod p) for any positive integer a not divisible by p. Thus, if a number does not satisfy this condition, it cannot be prime.
Any two distinct Fermat numbers m and n with m > n are relatively prime. Proof. Let m and n be distinct Fermat numbers with m > n; and suppose that d > 0 is a common
Fermat Primes • Fermat numbers are numbers of the form • Fermat believed every Fermat number is prime. • The only known Fermat primes are F 0 =3, F 1 =5, F 2 =17, F 3 =257, and F 4 =65537 • F n is composite for 4 < n < 31, (for example F 5 =4294967297=641*6700417) but no one knows if there are infinitely many Fermat Primes. 22n 1
For this lecture, before we get into Fermat’s Theorem, we’ll go over a few things: 1) Standard way to determine if a positive integer is prime or not. 2) Fundamental Theorem of Arithmetic 3) How to Prime Factorize an Integer (by hand and with code) 4) How to list all prime numbers upto a target integer n efficiently, in code. Standard ...
