65,537
65537 is the integer after 65536 and before 65538. In mathematics. 65537 is the largest known prime number of the form formula_1 (formula_2). 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 formula_3 formula_4 formula_5 formula_6 formula_7 In 1732, Leonhard Euler found that the next Fermat number is composite: formula_8 In 1880, showed that formula_9 65537 is  also the 17th Jacobsthal–Lucas number, and currently the largest known integer "n" for which the number formula_10 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". This value was used in RSA mainly for historical reasons; early raw RSA implementations (without proper padding) were vulnerable to very small exponents, while use of high exponents was computationally expensive with no advantage to security (assuming proper padding). 65537 is also used as the modulus in some Lehmer random number  generators, such as the one used by ZX Spectrum, which ensures that any seed value will be coprime to it (vital to ensure the maximum period) while also allowing efficient reduction by the modulus using a bit shift and subtract.