Fermat Number

A Fermat number is a positive integer of the form two raised to a power of two, itself raised to some exponent, plus one, written Fn for the exponent n, so that the first few Fermat numbers are three, five, seventeen, two hundred fifty-seven, and sixty-five thousand five hundred thirty-seven. They are named after the French mathematician Pierre de Fermat, who conjectured in the seventeenth century that every number of this form is prime, a claim that holds for the first five Fermat numbers but was disproved in 1732 by Leonhard Euler, who showed that the sixth one factors into smaller primes. No further prime Fermat numbers have ever been found beyond the first five, despite extensive computer searches of much larger candidates, and it remains an open question whether any more exist at all. Fermat numbers are also connected to geometry through a theorem of Carl Friedrich Gauss, who showed that a regular polygon can be constructed using only a compass and straightedge if its number of sides is a power of two multiplied by any number of distinct prime Fermat numbers.