31 (number)

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][7] as well as both a lucky prime^[8] and a happy number^[9] like its dual permutable prime in decimal (13).^[10]

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).^[11][12]

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.^[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).^[14]

31 is the 11th and final consecutive supersingular prime.^[15] After 31, the only supersingular primes are 41, 47, 59, and 71.

31 is the first prime centered pentagonal number,^[16] the fifth centered triangular number,^[17] and the first non-trivial centered decagonal number.^[18]

For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.^[19]

At 31, the Mertens function sets a new low of −4, a value which is not subceded until 110.^[20]

31 is a repdigit in base 2 (11111) and in base 5 (111).

The cube root of 31 is the value of π correct to four significant figures:

${\displaystyle {\sqrt[{3}]{3}}1=3.141\;{\color {red}38065\;\ldots }}$

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 ${\displaystyle 0}$);^[21] the partial sum of digits up to this point is ${\displaystyle 155=31\times 5.}$^[22] 31 is also the prime partial sum of digits of the decimal expansion of pi after the eighth digit.^[23][a]

The first five Euclid numbers of the form p₁ × p₂ × p₃ × ... × p_n + 1 (with p_n the nth prime) are prime:^[25]

• 3 = 2 + 1
• 7 = 2 × 3 + 1
• 31 = 2 × 3 × 5 + 1
• 211 = 2 × 3 × 5 × 7 + 1 and
• 2311 = 2 × 3 × 5 × 7 × 11 + 1

The following term, 30031 = 59 × 509 = 2 × 3 × 5 × 7 × 11 × 13 + 1, is composite.^[b] 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.^[26]

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.^[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). (11, 13) form the third twin prime pair^[6] between the fifth and sixth prime numbers whose indices add to 11, itself the prime index of 31.^[28] Where 31 is the prime index of the fourth Mersenne prime,^[2] the first three Mersenne primes (3, 7, 31) sum to the thirteenth prime number, 41.^[28][c] 13 and 31 are also the smallest values to reach record lows in the Mertens function, of −3 and −4 respectively.^[30]

The numbers 31, 331, 3331, 33331, 333331, 3333331, and 33333331 are all prime. For a time it was thought that every number of the form 3_w1 would be prime. However, the next nine numbers of the sequence are composite; their factorisations are:

• 333333331 = 17 × 19607843
• 3333333331 = 673 × 4952947
• 33333333331 = 307 × 108577633
• 333333333331 = 19 × 83 × 211371803
• 3333333333331 = 523 × 3049 × 2090353
• 33333333333331 = 607 × 1511 × 1997 × 18199
• 333333333333331 = 181 × 1841620626151
• 3333333333333331 = 199 × 16750418760469 and
• 33333333333333331 = 31 × 1499 × 717324094199.

The next term (3₁₇1) is prime, and the recurrence of the factor 31 in the last composite member of the sequence above can be used to prove that no sequence of the type R_wE or ER_w can consist only of primes, because every prime in the sequence will periodically divide further numbers.^{[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.^[31] It is also equal to the sum of the maximum number of areas generated by the first five n-sided polygons: 1, 2, 4, 8, 16, and as such, 31 is the first member that diverges from twice the value of its previous member in the sequence, by 1.

Icosahedral symmetry contains a total of thirty-one axes of symmetry; six five-fold, ten three-fold, and fifteen two-fold.^[32]

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

Thirty-one is also a slang term for masturbation in Turkish.^[33]

Look up thirty-one in Wiktionary, the free dictionary.

Wikimedia Commons has media related to 31 (number).

• Prime Curios! 31 from the Prime Pages