
Glossary: Prime Pages: Top 5000: 
GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) Euler's Theorem states that if gcd(a,n) = 1, then a^{phi(n)} = 1 (mod n). Here phi(n) (usually written ) is Euler's totient function: the number of integers in {1, 2, . . ., n1} which are relatively prime to n. When n is a prime, this theorem is just Fermat's little theorem. For example, phi(12)=4, so if gcd(a,12) = 1, then a^{4} = 1 (mod 12). The set of residue classes {d mod n  gcd(d,n)=1} modulo n form a multiplicative group, so Euler's theorem is a special case of Lagrange's theorem: the order of an element divides the order of a group.
Chris K. Caldwell © 19992020 (all rights reserved)
