For most of the last 500 years, the largest known prime has been a Mersenne prime, a number of the form 2p – 1 where p is itself prime. Such numbers are not always prime. For example, 211 − 1 = 2047 = 23 × 89.
Euclid proved circa 300 BC that if M is Mersenne prime, then M(M + 1)/2 is a perfect number, i.e. it is equal to the sum of the divisors less than itself. Euler proved two millennia later that all even perfect numbers must have this form. Since no odd perfect numbers have been discovered, the discovery of a new Mersenne prime implies the discovery of a new perfect number.
Using the same methods as in the posts from last year, we can say some things about how P appears in various bases.
In binary, P is written as a string of 74,207,281 ones.
In base four, P is a 1 followed by 37,103,640 threes.
In base eight, P is a 1 followed by 24,735,760 sevens.
In base 16, P is a 1 followed by 18,551,820 F’s.
In base 10, P has 22,338,618 digits, the first of which is 3 and the last of which is 1.