Difference between revisions of "Fermat number"
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Latest revision as of 18:18, 25 August 2009
A Fermat number is a number of the form where
is a nonnegative integer.
The first five Fermat numbers (for ) are
A prime Fermat number is known as a Fermat prime. Each of the first five Fermat numbers is a Fermat prime. Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime. However, this fails for
:
. In fact, the primes listed above are the only Fermat numbers known to be prime. The Fermat number
is known to be composite for
.
Relative primality of Fermat numbers
The Fermat numbers and
are relatively prime for all
Proof
We prove that the Fermat numbers satisfy the following recursion, from which the claimed result will follow: for all ,
We proceed by induction. For we have
, as desired. Suppose that
. Now observe that
as desired.
It follows that if that
, so
and
are relatively prime, as claimed.
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