# 31 (number)

 ← 30 31 32 →
Cardinal thirty-one
Ordinal 31st
(thirty-first)
Factorization prime
Divisors 1, 31
Roman numeral XXXI
Binary 111112
Ternary 10113
Quaternary 1334
Quinary 1115
Senary 516
Octal 378
Duodecimal 2712
Vigesimal 1B20
Base 36 V36

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

## In mathematics

Thirty-one is the third Mersenne prime (25 − 1) as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496, since 496 = 2(5 − 1)(25 − 1). 31 is the eighth Mersenne prime exponent. 31 is also the 4th lucky prime and the 11th supersingular prime.

31 is a centered triangular number, the lowest prime centered pentagonal number and a centered decagonal number.

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

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

No integer added up to its base 10 digits results in 31, making 31 a self number.

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

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 recurrence of the factor 31 in the last number 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. Here, 31 divides every fifteenth number in 3w1 (and 331 every 110th).

## In sports

The jersey number 31 has been retired by several North American sports teams in honor of past playing greats:

NASCAR driver Jeff Burton drives #31, a car which was subject to a controversy when one of the sponsors changed its name after merging with another company.

In ice hockey goaltenders often wear the number 31.

In football the number 31 has been retired by Queens Park Rangers F.C. in memory of Ray Jones (footballer born 1988)

## In other fields

Thirty-one is also: