Math Is Fun Forum

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Fermat numbers

A number of the form

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where n is a nonzero integer is called a Fermat number. The first five Fermat numbers F[sub]0[/sub], F[sub]1[/sub], F[sub]2[/sub], F[sub]3[/sub], F[sub]4[/sub] are primes. Pierre de Fermat conjectured that all such numbers were primes. This turned out to be false: F[sub]5[/sub] is a composite number, as Leonhard Euler discovered. It is not difficult to show that 641 divides 2[sup]32[/sup]+1.

It is also fairly easy to prove that any two distinct Fermat numbers are relatively prime. Fermat numbers have applications in geometry. Carl Friedrich Gauß showed that a regular polygon with n sides is constructible by ruler and compass if and only if n is a product of a power of 2 and distinct prime Fermat numbers.

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Last edited by JaneFairfax (2009-03-29 13:07:27)

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 46,328

Re: Fermat numbers

I remember this being taught at School, when the teacher was explaining 'Proof by Mathematical Induction'. The idea was to drive home the point that what may be true for the first few numbers need not necessarily be true for the entire set of counting numbers.

However, although I have read the book 'Fermat's Last Theorem' by Simon Singh, twice fully, and if reading in bits is taken into account, thrice, I don't remember reading this in the book. I am not too sure though. One good reason, my memory is not what it was 10 years back.

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