31st Pursuit Squadron

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

In 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. It is a lucky prime and a happy number; two properties it shares with 13, which is its dual emirp and permutable prime. 31 is also a primorial prime, like its twin prime, 29.

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 2^2''n'' + 1.

31 is the third Mersenne prime of the form 2^''n'' − 1. It is also the eighth Mersenne prime exponent, specifically for the number 2,147,483,647, which is the maximum positive value for a 32-bit signed binary integer in computing. After 3, it is the second Mersenne prime not to be a double Mersenne prime. 127, which is the 31st prime number, is a double Mersenne prime. The 31st triangular number is the perfect number 496, of the form 2^{(5 − 1)}(2⁵ − 1).

31 is a centered triangular number, the first 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.

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

The cube root of 31 is the value of pi correct to four significant figures.

The numbers 31, 331, 3331, , , , and 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:

* = 17 ×
* = 673 ×
* = 307 ×
* = 19 × 83 ×
* = 523 × 3049 ×
* = 607 × 1511 × 1997 ×
* = 181 ×
* = 199 × and
* = 31 × 1499 × .

The recurrence of the factor 31 in the last number 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. Here, 31 divides every fifteenth number in 3_w1 (and 331 every 110th).

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

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. 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.

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 Cataloguebr>object
NGC 31, a spiral galaxy

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