Last year I wrote a couple posts about what was then the largest known prime, 2^{57885161} – 1. Now there’s a new record, *P* = 2^{74207281} – 1.

For most of the last 500 years, the largest known prime has been a Mersenne prime, a number of the form 2^{p} – 1 where *p* is itself prime. Such numbers are not always prime. For example, 2^{11} − 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.

**Related posts**:

How can the last digit be 2, in base 10? Wouldn’t that make it divisible by 2 and therefore not prime?

Interesting note.

Your formula for perfect numbers is off. Wikipedia says (changing notation to be able to type more easily)

2^(p−1) * (2^p − 1)

Since 7 is a mersenne prime your formula would make 7*8 a perfect number when really it is 28 = 4 * 7 = 1 + 2 + 4 + 7 + 14

Thanks. I was missing a “/2”.