1000100000010000

The sequence 1000100000010000 is a string of binary digits. In the decimal system, this sequence represents the number 65537. This number is significant in mathematics as it is a Fermat number. Fermat numbers are of the form 2^(2^n) + 1. For n=4, the value is 2^(2^4) + 1, which simplifies to 2^16 + 1, resulting in 65536 + 1 = 65537.

Fermat numbers have been studied for their properties related to prime numbers. The first few Fermat numbers are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537. Pierre de Fermat conjectured that all numbers of this form are prime. Indeed, F0 through F4 are all prime numbers. However, F5, which is 2^(2^5) + 1 or 4294967297, was shown to be composite by Leonhard Euler. It is currently unknown whether any Fermat numbers beyond F4 are prime.

The number 65537 is the largest known Fermat prime. Its primality was proven by Fermat himself. It is also a constructible number, meaning that a regular polygon with 65537 sides can be constructed using only a compass and straightedge. This was a significant geometric discovery linked to the properties of Fermat primes. The binary representation 1000100000010000 is a direct way to express this important mathematical value.