65537

This number is a prime.

The largest known Fermat prime (2²⁴ + 1).

Just a small proportion of regular polygons (n-gons) can be constructed with compass and straightedge. Gauss proved that if n is a Fermat prime, then it is possible to construct an n-gon. Wantzel later proved this condition was also necessary (for prime n-gons), so the 65537-gon is currently the largest known constructible prime n-gon. It took Hermes 10 years and a 200-page manuscript to write down a procedure for its construction. Would you like to attempt it?

The smallest prime that is the sum of a nonzero square and a nonzero cube in four different ways: 65537 = 122² + 37³ = 219² + 26³ = 255² + 8³ = 256² + 1³. [Post]

To remember the digits of 65537, recite the following mnemonic: "Fermat prime, maybe the largest." Then count the number of letters in each word. [Brent]

Largest known prime mean of a Fermat prime and a Mersenne prime 65537 = (3+131071)/2. Richard Mathar has searched through all means that can be created from the existing values of the two OEIS sequences, Mersenne primes and Fermat primes. [Post]

65537 = (1/3)(196884 - 196560/(3*240)) where 196884 is the first coefficient in the j-invariant responsible for 'monstrous moonshine' and 196560 is the kissing number for the Leech lattice. The number 240 is the number of vectors in the E_8 root system. [Thomas]

The largest non-titanic prime of form n^(2*n) + 1, (case n = 4). [Loungrides]

The smallest octavan prime (form x^8 + y^8), where x and y are distinct non-prime digits, i.e., 1^8 + 4^8 = 65537. [Loungrides]

The largest known prime with a Hamming Weight of 2. In other words, the binary representation of 65537 has only 2 ones, which makes computer operations much easier. This fact is extremely important to cryptographers, who use 65537 as a common encryption exponent. [Price]

The 5th Fermat prime is also the sum of 97 consecutive primes, from 367 to 1009. [Rivera]

The smallest prime factor of 10^32768 + 1. [Perrenoud]

In cryptography, a popular method for generating prime numbers is to construct a candidate of the form: n = k × M + 65537a (mod M), where k, a are integers such that a is chosen randomly, k is a buffer used to ensure n is of the correct bit size and M is a primorial (a product of the first t successive primes).