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单词 FermatNumbers
释义

Fermat numbers


The n-th Fermat number is defined as:

Fn=22n+1.

Fermat incorrectly conjectured that all these numbers were primes,although he had no proof.The first 5 Fermat numbers: 3,5,17,257,65537 (corresponding to n=0,1,2,3,4) are all primes (so called Fermat primes)Euler was the first to point out the falsity of Fermat’s conjectureby proving that 641 is a divisorMathworldPlanetmathPlanetmath of F5. (In fact, F5=641×6700417).Moreover, no other Fermat number is known to be prime for n>4, so now it is conjectured that those are all prime Fermat numbers. It is also unknown whether there are infinitely many composite Fermat numbers or not.

One of the famous achievements of Gauss was to prove that the regular polygonMathworldPlanetmath of m sides can be constructed with ruler and compass if and only if m can be written as

m=2kFr1Fr2Frt

where k0 and the other factors are distinct primes of the form Fn (of course, t may be 0 here, i.e. m=2k is allowed).

There are many interesting properties involving Fermat numbers. For instance:

Fm=F0F1Fm-1+2

for any m1, which implies that Fm-2 is divisible by all smaller Fermat numbers.

The previous formula holds because

Fm-2=(22m+1)-2=22m-1=(22m-1-1)(22m-1+1)=(22m-1-1)Fm-1

and expanding recursively the left factor in the last expression gives the desired result.


References.
Krízek, Luca, Somer. 17 Lectures on Fermat Numbers. CMS Books in Mathematics.

TitleFermat numbers
Canonical nameFermatNumbers
Date of creation2013-03-22 11:42:46
Last modified on2013-03-22 11:42:46
Ownerdrini (3)
Last modified bydrini (3)
Numerical id30
Authordrini (3)
Entry typeDefinition
Classificationmsc 81T45
Classificationmsc 81T13
Classificationmsc 11A51
Classificationmsc 20L05
Classificationmsc 46L87
Classificationmsc 43A35
Classificationmsc 43A25
Classificationmsc 22D25
Classificationmsc 55U40
Classificationmsc 18B40
Classificationmsc 46L05
Classificationmsc 22A22
Classificationmsc 81R50
Classificationmsc 55U35
DefinesFermat prime
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