a3d7a1b4-582e-473b-9b54-1a6b20990495
prompter
Hello! Can you help me identify a prime number? 
115792089237316195423570985008687907853269984665640564039457584007908834671663
This prime is used for something and I'd like to know more about it, feel free to add details/trivia/factoids of any attributes you recognize.
7d572964-134f-4fa3-bc26-85d860a240f3
a3d7a1b4-582e-473b-9b54-1a6b20990495
assistant
A prime number is any whole number that can only be divided by itself or 1 to produce a whole number result, except 1 itself (although 1 used to be considered a prime number).
The specific prime number you gave is the Fermat prime of order 12, the largest known Fermat prime. Fermat primes are equal to 2^(2^n)+1.
It was discovered in 1996 by a computer program called Proth.exe, written by Yves Gallot.
It is now used in elliptic curve cryptography, for securing online transactions.
a638cb09-77a7-4488-a7a9-8c2ac814cf49
7d572964-134f-4fa3-bc26-85d860a240f3
prompter
Why is it called A Fermat prime and where did that equation come from?
c7ad5acf-5495-4770-812e-3029f8b5670b
a638cb09-77a7-4488-a7a9-8c2ac814cf49
assistant
The term "Fermat prime" is named after the French mathematician Pierre de Fermat, who studied prime numbers in the 17th century. Fermat conjectured that all numbers of the form 2^(2^n) + 1 are prime numbers, which are now known as Fermat numbers. However, this conjecture was later proven to be false when Leonhard Euler discovered that 2^32 + 1 is composite, disproving Fermat's conjecture.

The specific equation 2^(2^n) + 1 that defines Fermat primes comes from Fermat's observation that the first five numbers of this form, when n=0, 1, 2, 3, and 4, are prime numbers: 3, 5, 17, 257, and 65537, respectively. However, the next number in this sequence, when n=5, is not a prime number, and in fact, no other Fermat numbers are known to be prime for larger values of n.

Despite Fermat's conjecture being proven false, the search for Fermat primes still continues, as they have important applications in number theory and cryptography.