# 37 (number)

37 (thirty-seven) is the natural number following 36 and preceding 38.

## In mathematics

37 is the 12th prime number, and the 3rd isolated prime without a twin prime.
* 37 is the third star number and the fourth centered hexagonal number.
* The sum of the squares of the first 37 primes is divisible by 37.
* 37 is the median value for the second prime factor of an integer.
* Every positive integer is the sum of at most 37 fifth powers (see Waring's problem).
* It is the third cuban prime following 7 and 19.
* 37 is the fifth Padovan prime, after the first four prime numbers 2, 3, 5, and 7.
* It is the fifth lucky prime, after 3, 7, 13, and 31.
* 37 is a sexy prime, being 6 more than 31, and 6 less than 43.
* 37 remains prime when its digits are reversed, thus it is also a permutable prime.
37 is the first irregular prime with irregularity index of 1, where the smallest prime number with an irregularity index of 2 is the thirty-seventh prime number, 157.
The smallest magic square, using only primes and 1, contains 37 as the value of its central cell:
| 31 | 73 | 7 |
| 13 | 37 | 61 |
| 67 | 1 | 43 |
Its magic constant is 37 x 3 = 111, where 3 and 37 are the first and third base-ten unique primes (the second such prime is 11).
37 requires twenty-one steps to return to 1 in the 3x + 1 Collatz problem, as do adjacent numbers 36 and 38. The two closest numbers to cycle through the elementary {16, 8, 4, 2, 1} Collatz pathway are 5 and 32, whose sum is 37; also, the trajectories for 3 and 21 both require seven steps to reach 1. On the other hand, the first two integers that return $0$ for the Mertens function (2 and 39) have a difference of 37, where their product (2 × 39) is the twelfth triangular number 78. Meanwhile, their sum is 41, which is the constant term in Euler's lucky numbers that yield prime numbers of the form k$^2$ − k + 41, the largest of which (1601) is a difference of 78 (the twelfth triangular number) from the second-largest prime (1523) generated by this quadratic polynomial.
In moonshine theory, whereas all p ⩾ 73 are non-supersingular primes, the smallest such prime is 37.
37 is the sixth floor of imaginary parts of non-trivial zeroes in the Riemann zeta function. It is in equivalence with the sum of ceilings of the first two such zeroes, 15 and 22.
The secretary problem is also known as the 37% rule by ${\tfrac {1}{e}}\approx 37\%$.

### Decimal properties

For a three-digit number that is divisible by 37, a rule of divisibility is that another divisible by 37 can be generated by transferring first digit onto the end of a number. For example: 37|148 ➜ 37|481 ➜ 37|814. Any multiple of 37 can be mirrored and spaced with a zero each for another multiple of 37. For example, 37 and 703, 74 and 407, and 518 and 80105 are all multiples of 37; any multiple of 37 with a three-digit repdigit inserted generates another multiple of 37 (for example, 30007, 31117, 74, 70004 and 78884 are all multiples of 37).
Every equal-interval number (e.g. 123, 135, 753) duplicated to a palindrome (e.g. 123321, 753357) renders a multiple of both 11 and 111 (3 × 37 in decimal).
In decimal 37 is a permutable prime with 73, which is the twenty-first prime number. By extension, the mirroring of their digits and prime indexes makes 73 the only Sheldon prime.

### Geometric properties

There are precisely 37 complex reflection groups.
In three-dimensional space, the most uniform solids are:
* the five Platonic solids (with one type of regular face)
* the fifteen Archimedean solids (counting enantimorphs, all with multiple regular faces); and
* the sphere (with only a singular facet).
In total, these number twenty-one figures, which when including their dual polytopes (i.e. an extra tetrahedron, and another fifteen Catalan solids), the total becomes 6 + 30 + 1 = 37 (the sphere does not have a dual figure).
The sphere in particular circumscribes all the above regular and semiregular polyhedra (as a fundamental property); all of these solids also have unique representations as spherical polyhedra, or spherical tilings.

## Astronomy

* NGC 2169 is known as the 37 Cluster, due to its resemblance of the numerals.