← 30 31 32 →
Cardinalthirty-one
Ordinal31st
(thirty-first)
Factorizationprime
Prime11th
Divisors1, 31
Greek numeralΛΑ´
Roman numeralXXXI
Binary111112
Ternary10113
Senary516
Octal378
Duodecimal2712

31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.

## Mathematics

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]

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 a 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 π correct to four significant figures:

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

The first five Euclid numbers of the form p1 × p2 × p3 × ... × pn + 1 (with pn the nth prime) are prime:[20]

• 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.[a] 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.[21]

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][22] 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.[23] 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.[23][b] 13 and 31 are also the smallest values to reach record lows in the Mertens function, of −3 and −4 respectively.[25]

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 3w1 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 RwE or ERw 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.[26] 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.[27]

## In other fields

Thirty-one is also:

## Notes

1. ^ On the other hand, 13 is a largest p of a primorial prime of the form pn# − 1 = 30029 (sequence A057704 in the OEIS).
2. ^ Also, the sum between the sum and product of the first two Mersenne primes is (3 + 7) + (3 × 7) = 10 + 21 = 31, where its difference (11) is the prime index of 31.[23] Thirty-one is also in equivalence with 14 + 17, which are respectively the seventh composite[24] and prime numbers,[23] whose difference in turn is three.

## References

1. ^ "Sloane's A003052 : Self numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
2. ^ a b Sloane, N. J. A. (ed.). "Sequence A000668 (Mersenne primes (primes of the form 2^n - 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-07.
3. ^ Sloane, N. J. A. (ed.). "Sequence A000043 (Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-07.
4. ^ Sloane, N. J. A. (ed.). "Sequence A077586 (Double Mersenne primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-07.
5. ^ "Sloane's A000217 : Triangular numbers". The On-Line Encyclopedia oof Integer Sequences. OEIS Foundation. Retrieved 2022-09-30.
6. ^ a b Sloane, N. J. A. (ed.). "Sequence A228486 (Near primorial primes: primes p such that p+1 or p-1 is a primorial number (A002110))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-07.
7. ^ Sloane, N. J. A. (ed.). "Sequence A077800 (List of twin primes {p, p+2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-07.
8. ^ "Sloane's A031157 : Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
9. ^ "Sloane's A007770 : Happy numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
10. ^ a b Sloane, N. J. A. (ed.). "Sequence A003459 (Absolute primes (or permutable primes): every permutation of the digits is a prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-07.
11. ^ Conway, John H.; Guy, Richard K. (1996). "The Primacy of Primes". The Book of Numbers. New York, NY: Copernicus (Springer). pp. 137–142. doi:10.1007/978-1-4612-4072-3. ISBN 978-1-4612-8488-8. OCLC 32854557. S2CID 115239655.
12. ^ Sloane, N. J. A. (ed.). "Sequence A004729 (... the 31 regular polygons with an odd number of sides constructible with ruler and compass)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26.
13. ^ Sloane, N. J. A. (ed.). "Sequence A000203 (The sum of the divisors of n. Also called sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-23.
14. ^ "Sloane's A002267 : The 15 supersingular primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
15. ^ "Sloane's A005891 : Centered pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
16. ^ "Sloane's A005448 : Centered triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
17. ^ "Sloane's A062786 : Centered 10-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
18. ^ Hwang, Frank. (1992). The Steiner tree problem. Richards, Dana, 1955-, Winter, Pawel, 1952-. Amsterdam: North-Holland. p. 14. ISBN 978-0-444-89098-6. OCLC 316565524.
19. ^ Sloane, N. J. A. (ed.). "Sequence A002321 (Mertens's function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-07.
20. ^ Sloane, N. J. A. (ed.). "Sequence A006862 (Euclid numbers: 1 + product of the first n primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-01.
21. ^ Conway, John H.; Guy, Richard K. (1996). "The Primacy of Primes". The Book of Numbers. New York, NY: Copernicus (Springer). pp. 133–135. doi:10.1007/978-1-4612-4072-3. ISBN 978-1-4612-8488-8. OCLC 32854557. S2CID 115239655.
22. ^ Sloane, N. J. A. (ed.). "Sequence A006567 (Emirps (primes whose reversal is a different prime).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-16.
23. ^ a b c d Sloane, N. J. A. (ed.). "Sequence A00040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-09.
24. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x greater than 1 and y greater than 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-10.
25. ^ Sloane, N. J. A. (ed.). "Sequence A051402 (Inverse Mertens function: smallest k such that |M(k)| is n, where M(x) is Mertens's function A002321.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-08.
26. ^ "Sloane's A000127 : Maximal number of regions obtained by joining n points around a circle by straight lines". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-09-30.
27. ^ Hart, George W. (1998). "Icosahedral Constructions" (PDF). In Sarhangi, Reza (ed.). Bridges: Mathematical Connections in Art, Music, and Science. Proceedings of the Bridges Conference. Winfield, Kansas. p. 196. ISBN 978-0966520101. OCLC 59580549. S2CID 202679388.((cite book)): CS1 maint: location missing publisher (link)
28. ^ "Tureng - 31 çekmek - Türkçe İngilizce Sözlük". tureng.com. Retrieved 2023-01-18.