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]
37 requires twenty-one steps to return to 1 in the 3x + 1Collatz 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]
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.[18] 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).
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).
^Koninck, Jean-Marie de; Koninck, Jean-Marie de (2009). Those fascinating numbers. Providence, R.I: American Mathematical Society.
ISBN978-0-8218-4807-4.
^Weisstein, Eric W.
"Waring's Problem". mathworld.wolfram.com. Retrieved 2020-08-21.