# 37 (number)

Short description: Natural number
 ← 36 37 38 →
Cardinalthirty-seven
Ordinal37th
(thirty-seventh)
Factorizationprime
Prime12th
Divisors1, 37
Greek numeralΛΖ´
Roman numeralXXXVII
Binary1001012
Ternary11013
Quaternary2114
Quinary1225
Senary1016
Octal458
Duodecimal3112
Vigesimal1H20
Base 361136

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.[2]
• The sum of the squares of the first 37 primes is divisible by 37.[3]
• 37 is the median value for the second prime factor of an integer.[4]
• Every positive integer is the sum of at most 37 fifth powers (see Waring's problem).[5]
• It is the third cuban prime following 7 and 19.[6]
• 37 is the fifth Padovan prime, after the first four prime numbers 2, 3, 5, and 7.[7]
• It is also the fifth lucky prime, after 3, 7, 13, and 31.[8]
• 37 is the third star number[9] and the fourth centered hexagonal number.[10]

There are exactly 37 complex reflection groups.

The smallest magic square, using only primes and 1, contains 37 as the value of its central cell:[11]

 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).[12]

In decimal 37 is a permutable prime with 73, which is the 21st prime number. By extension, the mirroring of their digits and prime indexes makes 73 the only Sheldon prime. In moonshine theory, whereas all p ⩾ 73 are non-supersingular primes, the smallest such prime is 37.

37 requires twenty-one steps to return to 1 in the 3x + 1 Collatz problem, as do adjacent numbers 36 and 38.[13] The two closest numbers to cycle through the elementary {16, 8, 4, 2, 1} Collatz pathway are 5 and 32, whose sum is 37.[14] The trajectories for 3 and 21 both require seven steps to reach 1.[13]

The first two integers that return $\displaystyle{ 0 }$ for the Mertens function (2 and 39) have a difference of 37.[15] Their product (2 × 39) is the twelfth triangular number 78. 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 from the second-largest prime (1523) generated by this quadratic polynomial.[16]

### In decimal

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.[17]

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 science

### Astronomy

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

## In other fields

House number in Baarle (in its Belgian part)

Thirty-seven is:

• The number of the French department Indre-et-Loire[18]
• The number of slots in European roulette (numbered 0 to 36, the 00 is not used in European roulette as it is in American roulette)
• The RSA public exponent used by PuTTY
• Richard Nixon, 37th president of the United States .
• DEVO song "37" from "Hardcore Devo: Volume Two"

• List of highways numbered 37
• Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania
• I37 (disambiguation)

## 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.)". OEIS Foundation. Retrieved 2022-12-05.
2. "Sloane's A000928: Irregular primes". OEIS Foundation.
3. 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)". OEIS Foundation. Retrieved 2022-06-02.
4. Koninck, Jean-Marie de; Koninck, Jean-Marie de (2009). Those fascinating numbers. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-4807-4.
5. Weisstein, Eric W.. "Waring's Problem" (in en).
6. "Sloane's A002407: Cuban primes". OEIS Foundation.
7. "Sloane's A000931: Padovan sequence". OEIS Foundation.
8. "Sloane's A003215: Hex (or centered hexagonal) numbers". OEIS Foundation.
9. Henry E. Dudeney (1917). Amusements in Mathematics. London: Thomas Nelson & Sons, Ltd.. p. 125. ISBN 978-1153585316. OCLC 645667320.
10. "Sloane's A040017: Unique period primes". OEIS Foundation.
11. 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.)". OEIS Foundation. Retrieved 2023-09-18.
12. Sloane, N. J. A.. "3x+1 problem". The OEIS Foundation.
13. Sloane, N. J. A., ed. "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". OEIS Foundation. Retrieved 2023-09-02.
14. 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.)". OEIS Foundation. Retrieved 2023-09-02.
15. Vukosav, Milica (2012-03-13). "NEKA SVOJSTVA BROJA 37" (in hr). Matka: Časopis za Mlade Matematičare 20 (79): 164. ISSN 1330-1047.