
Glossary: Prime Pages: Top 5000: 
GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) Only five Fermat primes are known, and the Fermat numbers grow so quickly that it may be years before the first undecided case: F_{31} = is shown prime or compositeunless we luck onto a divisor. Ever since Euler found the first Fermat divisor (divisor of a Fermat composite), factorers have been collecting these rare numbers. (Luck has prevailed! On 12 April 2001, Alexander Kruppa found that 46931635677864055013377 divides F_{31}, so now F_{33} is the least Fermat with unknown status!) Euler showed that every divisor of F_{n} (n greater than 2) must have the form k^{.}2^{n+2}+1 for some integer k. For this reason, when we find a large prime of the form k^{.}2^{n}+1 (with k small), we usually check to see if it divides a Fermat number. The probability of the number k^{.}2^{n}+1 dividing any Fermat number appears to be 1/k.
See Also: Fermats, CunninghamProject, FermatQuotient Related pages (outside of this work)
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Chris K. Caldwell © 19992019 (all rights reserved)
