37 is the firstirregular prime with irregularity index of1,[10] where the smallest prime number with an irregularity index of2 is the thirty-seventh prime number,157.[11]
37 requires twenty-one steps to return to 1 in the3x + 1Collatz problem, as do adjacent numbers36 and38.[14] The two closest numbers to cycle through the elementary {16, 8, 4, 2, 1} Collatz pathway are5 and32, whose sum is 37;[15] also, the trajectories for3 and21 both require seven steps to reach 1.[14] On the other hand, the first twointegers that return for theMertens function (2 and39) have adifference of 37,[16] where their product (2 × 39) is the twelfthtriangular number 78. Meanwhile, their sum is41, which is the constant term inEuler's lucky numbers that yield prime numbers of the formk2 −k + 41, the largest of which (1601) is a difference of78 (the twelfthtriangular number) from the second-largest prime (1523) generated by thisquadratic polynomial.[17]
For a three-digit number that is divisible by 37, arule 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-digitrepdigit 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 total, these number twenty-one figures, which when including theirdual polytopes (i.e. an extratetrahedron, and another fifteenCatalan 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. Retrieved2020-08-21.
