Pépin’s theorem

Theorem (Pépin). A Fermat number Fn=22n+1 is prime only if


In other words, if 3 raised to the largest power of two not greater than the Fermat number leaves as a remainder the next higher power of two when divided by that Fermat number (since Fn-1=22n), then that Fermat number is a Fermat prime.

For example, 17=222+1 is a Fermat prime, and we can see that 38=6561, which leaves a remainder of 16 when divided by 17. The smallest Fermat number not to be a prime is 4294967297, as it is the product of 641 and 6700417, and 32147483648 divided by 4294967297 leaves a remainder of 10324303 rather than 4294967296.

Title Pépin’s theorem
Canonical name PepinsTheorem
Date of creation 2013-03-22 18:53:09
Last modified on 2013-03-22 18:53:09
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Theorem
Classification msc 11A51
Synonym Pepin’s theorem