31 (number)

← 30 31 32 →
← 30 31 32 33 34 35 36 37 38 39 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	thirty-one
Ordinal	31st (thirty-first)
Factorization	prime
Prime	11th
Divisors	1, 31
Greek numeral	ΛΑ´
Roman numeral	XXXI
Binary	111112
Ternary	10113
Senary	516
Octal	378
Duodecimal	2712
Hexadecimal	1F16

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]}

31 is a centered pentagonal number.

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]

In science

• The atomic number of gallium

Astronomy

• Messier object M31, a magnitude 4.5 galaxy in the constellation Andromeda. It is also known as the Andromeda Galaxy, and is readily visible to the naked eye in a modestly dark sky.
• The New General Catalogue object NGC 31, a spiral galaxy in the constellation Phoenix

In sports

• Ice hockey goaltenders often wear the number 31.^{[ citation needed ]}

In other fields

Thirty-one is also:

• The number of days in each of the months January, March, May, July, August, October and December
• The number of the date that Halloween and New Year's Eve are celebrated
• The code for international direct-dial phone calls to the Netherlands
• Thirty-one, a card game
• The number of kings defeated by the incoming Israelites in Canaan according to Joshua 12:24: "all the kings, one and thirty" (Wycliffe Bible translation)
• A type of game played on a backgammon board
• The number of flavors of Baskin-Robbins ice cream; the shops are called 31 Ice Cream in Japan
• ISO 31 is the ISO's standard for quantities and units
• In the title of the anime Ulysses 31
• In the title of Nick Hornby's book 31 Songs
• A women's honorary at The University of Alabama (XXXI)^{[ citation needed ]}
• The number of the French department Haute-Garonne
• In music, 31-tone equal temperament is a historically significant tuning system (31 equal temperament), first theorized by Christiaan Huygens and promulgated in the 20th century by Adriaan Fokker
• Number of letters in Macedonian alphabet
• Number of letters in Ottoman alphabet ^{[ citation needed ]}
• A slang term for masturbation in Turkish. ^[33]

Notes

1. On the other hand, "31" as a string represents the first decimal expansion of pi truncated to numbers such that the partial sums of the decimal digits are square numbers. ^[24]
2. On the other hand, 13 is a largest p of a primorial prime of the form p_n# − 1 = 30029 (sequence A057704 in the OEIS).
3. 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. ^[28] Thirty-one is also in equivalence with 14 + 17, which are respectively the seventh composite ^[29] and prime numbers, ^[28] 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. Sloane, N. J. A. (ed.). "SequenceA000668(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.). "SequenceA000043(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.). "SequenceA077586(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. Sloane, N. J. A. (ed.). "SequenceA228486(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.). "SequenceA077800(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. Sloane, N. J. A. (ed.). "SequenceA003459(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.). "SequenceA004729(... 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.). "SequenceA000203(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, N. J. A. (ed.). "SequenceA001422(Numbers which are not the sum of distinct squares.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-08-03.
15. "Sloane's A002267 : The 15 supersingular primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
16. "Sloane's A005891 : Centered pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
17. "Sloane's A005448 : Centered triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
18. "Sloane's A062786 : Centered 10-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
19. 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.
20. Sloane, N. J. A. (ed.). "SequenceA002321(Mertens's function)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-07.
21. Sloane, N. J. A. (ed.). "SequenceA072136(Position of the first zero in the fractional part of the base n expansion of Pi.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-05-30.
22. Sloane, N. J. A. (ed.). "SequenceA046974(Partial sums of digits of decimal expansion of Pi.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
23. Sloane, N. J. A. (ed.). "SequenceA133213(Prime partial sums of digits of decimal expansion of pi (A000796).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-06-02.
24. Sloane, N. J. A. (ed.). "SequenceA276111(Decimal expansion of Pi truncated to numbers such that the partial sums of the decimal digits are perfect squares.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-06-02.
25. Sloane, N. J. A. (ed.). "SequenceA006862(Euclid numbers: 1 + product of the first n primes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-10-01.
26. 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.
27. Sloane, N. J. A. (ed.). "SequenceA006567(Emirps (primes whose reversal is a different prime).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-16.
28. Sloane, N. J. A. (ed.). "SequenceA00040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-09.
29. Sloane, N. J. A. (ed.). "SequenceA002808(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.
30. Sloane, N. J. A. (ed.). "SequenceA051402(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.
31. "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.
32. 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.
33. "Tureng - 31 çekmek - Türkçe İngilizce Sözlük". tureng.com. Retrieved 2023-01-18.

External links

• Prime Curios! 31 from the Prime Pages