Theorem (Pépin). A Fermat number 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 ), then that Fermat number is a Fermat prime.
For example, is a Fermat prime, and we can see that , 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 divided by 4294967297 leaves a remainder of 10324303 rather than 4294967296.
|Date of creation||2013-03-22 18:53:09|
|Last modified on||2013-03-22 18:53:09|
|Last modified by||PrimeFan (13766)|