65,537

In mathematics

65537 is the largest known prime number of the form 2^{2^{n}} +1 (n = 4), and is most likely the last one. Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann Gustav Hermes gave the first explicit construction of this polygon. In number theory, primes of this form are known as Fermat primes, named after the mathematician Pierre de Fermat. The only known prime Fermat numbers are

2^{2^{0}} + 1 = 2^{1} + 1 = 3,

2^{2^{1}} + 1= 2^{2} +1 = 5,

2^{2^{2}} + 1 = 2^{4} +1 = 17,

2^{2^{3}} + 1= 2^{8} + 1= 257,

2^{2^{4}} + 1 = 2^{16} + 1 = 65537.

In 1732, Leonhard Euler found that the next Fermat number is composite:

2^{2^{5}} + 1 = 2^{32} + 1 = 4294967297 = 641 \times 6700417

In 1880, showed that

2^{2^{6}} + 1 = 2^{64} + 1 = 274177 \times 67280421310721

65537 is also the 17th Jacobsthal–Lucas number, and currently the largest known integer n for which the number 10^{n} + 27 is a probable prime.

Applications

65537 is commonly used as a public exponent in the RSA cryptosystem. Because it is the Fermat number with , the common shorthand is "F" or "F4".{{cite web | access-date = 2017-05-24 | archive-date = 2017-03-13 | archive-url = https://web.archive.org/web/20170313225729/https://www.openssl.org/docs/man1.0.2/apps/genrsa.html | url-status = dead

65537 is also used as the modulus in some Lehmer random number generators, such as the one used by ZX Spectrum,{{cite book |chapter-url=https://worldofspectrum.org/ZXBasicManual/zxmanchap11.html |access-date = 2022-05-26

References

References

1. (2017). "Expect at most one billionth of a new Fermat Prime!". The Mathematical Intelligencer.
2. Conway, J. H.. (1996). "The Book of Numbers". Springer-Verlag.
3. "Sequences by difficulty of search".
4. "RSA with small exponents?".