31 (number)

It is a superprime and a self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31.

[1] It is the third Mersenne prime of the form 2n − 1,[2] and the eighth Mersenne prime exponent,[3] in-turn yielding the maximum positive value for a 32-bit signed binary integer in computing: 2,147,483,647.

[13] In total, only thirty-one integers are not the sum of distinct squares (31 is the sixteenth such number, where the largest is 124).

The cube root of 31 is the value of π correct to four significant figures: The thirty-first digit in the fractional part of the decimal expansion for pi in base-10 is the last consecutive non-zero digit represented, starting from the beginning of the expansion (i.e, the thirty-second single-digit string is the first

[10][27] Meanwhile 1310 in ternary is 1113 and 3110 in quinary is 1115, with 1310 in quaternary represented as 314 and 3110 as 1334 (their mirror permutations 3314 and 134, equivalent to 61 and 7 in decimal, respectively, are also prime).

[28][c] 13 and 31 are also the smallest values to reach record lows in the Mertens function, of −3 and −4 respectively.

[citation needed] 31 is the maximum number of areas inside a circle created from the edges and diagonals of an inscribed six-sided polygon, per Moser's circle problem.

[32] 31 equal temperament is a popular microtonal tuning for musical instruments because it provides a good approximation of harmonic intervals.