Sequence A000215, Fermat Numbers
The Fermat numbers, Sloane's A000215, are numbers of the form 22m+1 for non-negative integer m.
The sequence begins: 220+1 = 3, 221+1 = 5, 222+1 = 17, 223+1 = 257, 224+1 = 65537, 225+1 = 4294967297, 226+1 = 18446744073709551617, 227+1 = 340282366920938463463374607431768211457, 228+1 = 115792089237316195423570985008687907853269984665640564039457584007913129639937, 229+1 = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097, ...
In 1640, Fermat knew that F0 through F4 are prime and conjectured that all higher Fermat numbers were also prime. In the years since the Fermat numbers and their factors have been the subject of much research.
In 1732, Euler showed that all factors of a Fermat number Fm must be of the form k×2m+1+1. This limits the number of primes that must be tested to factor a given Fermat number. It then becomes easy to find that F5 = 641×6700417, because 641 is only the 5th prime that satisfies the requirement of being equal to k×26+1 for some k.
In 1801, Gauss proved that a regular polygon can be constructed by the classical technique of straightedge and compass ("ruler and compass") if and only if the number of sides is a product of 2n and prime Fermat numbers. Thus, for example, a 17-sided polygon can be constructed, as can a 51-sided polygon (because 51=3×17) but a 7-sided or 21-sided polygon cannot because both are multiples of 7.
In 1878 Lucas improved on the Euler requirement by showing that a Fermat factor must be of the form k×2m+2+1. So for example, the factors of F5 are of the form k×27+1 (and 641 is the first prime that satisfies this requirement).
The factors of Fermat numbers are discussed here.
Some other sequences are discussed here.
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