| Number: | 31 |
| Factorization: | prime |
| Prime: | 11th |
| Divisor: | 1, 31 |
31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.
31 is the 11th prime 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. After 3, it is the second Mersenne prime not to be a double Mersenne prime, while the 31st prime number (127) is the second double Mersenne prime, following 7.[4] On the other hand, the thirty-first triangular number is the perfect number 496, of the form 2(5 − 1)(25 − 1) by the Euclid-Euler theorem.[5] 31 is also a primorial prime like its twin prime (29),[6] as well as both a lucky prime[7] and a happy number[8] like its dual permutable prime in decimal (13).[9]
31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge, from combinations of known Fermat primes of the form 22n + 1 (they are 3, 5, 17, 257 and 65537).[10] [11]
Only two numbers have a sum-of-divisors equal to 31: 16 (1 + 2 + 4 + 8 + 16) and 25 (1 + 5 + 25), respectively the square of 4, and of 5.[12] In total, only thirty-one integers are not the sum of distinct squares (31 is the sixteenth such number, where the largest is 124).[13]
31 is the 11th and final consecutive supersingular prime.[14] After 31, the only supersingular primes are 41, 47, 59, and 71.
31 is the first prime centered pentagonal number,[15] the fifth centered triangular number,[16] and the first non-trivial centered decagonal number.[17]
For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.[18]
At 31, the Mertens function sets a new low of −4, a value which is not subceded until 110.[19]
31 is a repdigit in base 2 (11111) and in base 5 (111).
The cube root of 31 is the value of correct to four significant figures:
\sqrt[3]31=3.141 {\color{red}38065 \ldots}
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
0
155=31 x 5.
The first five Euclid numbers of the form p1 × p2 × p3 × ... × pn + 1 (with pn the nth prime) are prime:[23]
The following term, 30031 = 59 × 509 = 2 × 3 × 5 × 7 × 11 × 13 + 1, is composite. The next prime number of this form has a largest prime p of 31: 2 × 3 × 5 × 7 × 11 × 13 × ... × 31 + 1 ≈ 8.2 × 1033.[24]
While 13 and 31 in base-ten are the proper first duo of two-digit permutable primes and emirps with distinct digits in base ten, 11 is the only two-digit permutable prime that is its own permutable prime.[25] 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). (11, 13) form the third twin prime pair between the fifth and sixth prime numbers whose indices add to 11, itself the prime index of 31.[26] Where 31 is the prime index of the fourth Mersenne prime, the first three Mersenne primes (3, 7, 31) sum to the thirteenth prime number, 41. 13 and 31 are also the smallest values to reach record lows in the Mertens function, of −3 and −4 respectively.[27]