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A000215 - Fermat numbers
A000215 - Fermat numbers: 2^(2^n) + 1, n >= 0
3, 5, 17, 257, 65537, 4294967297, ...
In 1650, Pierre de Fermat conjectured that numbers of the form Fn = 22n+1 were prime. But these numbers get large quickly (n=5 has ten digits), and it wasn't until 1732 that Euler factored F5 = 641 * 6700417. In 1855, n = 6 solved, and through the middle of the 20th century, several factors of other Fermat numbers were found.
When computers took over, this work happened faster, but still not quickly. As of today, only seven Fermat numbers are completely factored. The largest of those, F11 has only five prime factors, and one of those is 560 digits long, and none have been finished since 1990.
But there still several open questions. Are there any prime Fermat numbers for n > 4? If there are, are there an infinite number of them? Richard Guy and John Conway made a bet in 1996 that no Fermat numbers would be completely factored in the next 20 years -- and Conway seems likely to collect.
Related: A057755 - Number of digits in n-th Fermat number
