In mathematics
37 is the 12th prime number, and the 3rd isolated prime without a twin prime. [1]
- 37 is the third star number [2] and the fourth centered hexagonal number. [3]
- The sum of the squares of the first 37 primes is divisible by 37. [4]
- 37 is the median value for the second prime factor of an integer. [5]
- Every positive integer is the sum of at most 37 fifth powers (see Waring's problem). [6]
- It is the third cuban prime following 7 and 19. [7]
- 37 is the fifth Padovan prime, after the first four prime numbers 2, 3, 5, and 7. [8]
- It is the fifth lucky prime, after 3, 7, 13, and 31. [9]
- 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, [10] where the smallest prime number with an irregularity index of 2 is the thirty-seventh prime number, 157. [11]
The smallest magic square, using only primes and 1, contains 37 as the value of its central cell: [12]
| 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). [13]
37 requires twenty-one steps to return to 1 in the 3x + 1 Collatz problem, as do adjacent numbers 36 and 38. [14] The two closest numbers to cycle through the elementary {16, 8, 4, 2, 1} Collatz pathway are 5 and 32, whose sum is 37; [15] also, the trajectories for 3 and 21 both require seven steps to reach 1. [14] On the other hand, the first two integers that return for the Mertens function (2 and 39) have a difference of 37, [16] 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 k2 − 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. [17]
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. [18] It is in equivalence with the sum of ceilings of the first two such zeroes, 15 and 22. [19]
The secretary problem is also known as the 37% rule by .
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. [20] 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. [21]
