# 37 (number)

(Redirected from Thirty-seven)
 ← 36 37 38 →
Cardinalthirty-seven
Ordinal37th
(thirty-seventh)
Factorizationprime
Prime12th
Divisors1, 37
Greek numeralΛΖ´
Roman numeralXXXVII
Binary1001012
Ternary11013
Senary1016
Octal458
Duodecimal3112

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.[1]

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 ${\displaystyle 0}$ 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 k2k + 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.

The secretary problem is also known as the 37% rule by ${\displaystyle {\tfrac {1}{e}}\approx 37\%}$.

### 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.[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 repunit inserted generates another multiple of 37 (for example, 30007, 31117, 74, 70004 and 78884 are all multiples of 37).

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:

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.[19]

## In science

### Astronomy

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

Thirty-seven is:

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A007510 (Single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-05.
2. ^ "Sloane's A003154: Centered 12-gonal numbers. Also star numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
3. ^ "Sloane's A003215: Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
4. ^ Sloane, N. J. A. (ed.). "Sequence A111441 (Numbers k such that the sum of the squares of the first k primes is divisible by k)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-06-02.
5. ^ Koninck, Jean-Marie de; Koninck, Jean-Marie de (2009). Those fascinating numbers. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-4807-4.
6. ^ Weisstein, Eric W. "Waring's Problem". mathworld.wolfram.com. Retrieved 2020-08-21.
7. ^ "Sloane's A002407: Cuban primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
8. ^ "Sloane's A000931: Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
9. ^ "Sloane's A031157: Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
10. ^ "Sloane's A000928: Irregular primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
11. ^ Sloane, N. J. A. (ed.). "Sequence A073277 (Irregular primes with irregularity index two.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-03-25.
12. ^ Henry E. Dudeney (1917). Amusements in Mathematics (PDF). London: Thomas Nelson & Sons, Ltd. p. 125. ISBN 978-1153585316. OCLC 645667320. Archived (PDF) from the original on 2023-02-01.
13. ^ "Sloane's A040017: Unique period primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
14. ^ a b Sloane, N. J. A. (ed.). "Sequence A006577 (Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-18.
15. ^ Sloane, N. J. A. "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-09-18.
16. ^ Sloane, N. J. A. (ed.). "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
17. ^ Sloane, N. J. A. (ed.). "Sequence A196230 (Euler primes: values of x^2 - x + k for x equal to 1..k-1, where k is one of Euler's "lucky" numbers 2, 3, 5, 11, 17, 41.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
18. ^ Vukosav, Milica (2012-03-13). "NEKA SVOJSTVA BROJA 37". Matka: Časopis za Mlade Matematičare (in Croatian). 20 (79): 164. ISSN 1330-1047.
19. ^ Har'El, Zvi (1993). "Uniform Solution for Uniform Polyhedra" (PDF). Geometriae Dedicata. 47. Netherlands: Springer Publishing: 57–110. doi:10.1007/BF01263494. MR 1230107. S2CID 120995279. Zbl 0784.51020.
See, 2. THE FUNDAMENTAL SYSTEM.
20. ^
21. ^ Why is this number everywhere?. Retrieved 2024-03-29 – via www.youtube.com.