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#math trivia for #September8: #251 is the largest 8-bit prime. What are the largest 2-, 3-, 4-, 5-, 6- and 7-bit primes? (Why no 1-bits?)

— Burt Kaliski Jr. (@modulomathy) September 8, 2013

The largest primes of length 2 to 7 bits are 3, 7, 13, 31, 61, and 127.

As it turns out, the largest primes of length 2, 3, 5, and 7 bits are also the largest numbers of those particular lengths. This is because the primes are all Mersenne primes: primes of the form 2^p-1 where p is also prime.

Interestingly, when n is composite, 2ⁿ-1 is also composite.  In particular, if p is a divisor of n, then 2^p-1 is a divisor of 2ⁿ-1.   This is why the largest primes of length 4 and 6 are not also the largest numbers of those particular lengths.

It is not necessarily the case that when p is prime, 2^p-1 is also prime, though it works for the four examples here.

The reason that there are no one-bit primes is that the only 1-bit numbers are 0 (which could technically also be considered a 0-bit number) and 1. Neither is prime.  At one time, mathematicians considered 1 to be a prime, because, like other prime numbers, it is divisible only by 1 and itself. However, 1 is even more special: It is the multiplicative identity. For technical reasons, many properties involving prime numbers work out better if 1 keeps its special status.