37 (number)

From HandWiki
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
Hexadecimal2516
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 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 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 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 1e≈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.[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:

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]

Astronomy

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

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. https://oeis.org/A007510. Retrieved 2022-12-05. 
  2. ↑ "Sloane's A003154: Centered 12-gonal numbers. Also star numbers". OEIS Foundation. https://oeis.org/A003154. 
  3. ↑ "Sloane's A003215: Hex (or centered hexagonal) numbers". OEIS Foundation. https://oeis.org/A003215. 
  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)". OEIS Foundation. https://oeis.org/A111441. 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" (in en). https://mathworld.wolfram.com/WaringsProblem.html. 
  7. ↑ "Sloane's A002407: Cuban primes". OEIS Foundation. https://oeis.org/A002407. 
  8. ↑ "Sloane's A000931: Padovan sequence". OEIS Foundation. https://oeis.org/A000931. 
  9. ↑ "Sloane's A031157: Numbers that are both lucky and prime". OEIS Foundation. https://oeis.org/A031157. 
  10. ↑ "Sloane's A000928: Irregular primes". OEIS Foundation. https://oeis.org/A000928. 
  11. ↑ Sloane, N. J. A., ed. "Sequence A073277 (Irregular primes with irregularity index two.)". OEIS Foundation. https://oeis.org/A073277. Retrieved 2024-03-25. 
  12. ↑ Henry E. Dudeney (1917). Amusements in Mathematics. London: Thomas Nelson & Sons, Ltd.. p. 125. ISBN 978-1153585316. OCLC 645667320. http://djm.cc/library/Amusements_in_Mathematics_Dudeney_edited02.pdf. 
  13. ↑ "Sloane's A040017: Unique period primes". OEIS Foundation. https://oeis.org/A040017. 
  14. ↑ 14.0 14.1 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. https://oeis.org/A006577. Retrieved 2023-09-18. 
  15. ↑ Sloane, N. J. A.. "3x+1 problem". The OEIS Foundation. http://oeis.org/wiki/3x%2B1_problem. 
  16. ↑ Sloane, N. J. A., ed. "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". OEIS Foundation. https://oeis.org/A028442. 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.)". OEIS Foundation. https://oeis.org/A196230. Retrieved 2023-09-02. 
  18. ↑ Sloane, N. J. A., ed. "Sequence A013629 (Floor of imaginary parts of nontrivial zeros of Riemann zeta function.)". OEIS Foundation. https://oeis.org/A013629. 
  19. ↑ Sloane, N. J. A., ed. "Sequence A092783 (Ceiling of imaginary parts of nontrivial zeros of Riemann zeta function.)". OEIS Foundation. https://oeis.org/A092783. 
  20. ↑ Vukosav, Milica (2012-03-13). "NEKA SVOJSTVA BROJA 37" (in hr). Matka: Časopis za Mlade Matematičare 20 (79): 164. ISSN 1330-1047. https://hrcak.srce.hr/81042. 
  21. ↑ Har'El, Zvi (1993). "Uniform Solution for Uniform Polyhedra". Geometriae Dedicata (Netherlands: Springer Publishing) 47: 57–110. doi:10.1007/BF01263494. http://harel.org.il/zvi/docs/uniform.pdf. 
    See, 2. THE FUNDAMENTAL SYSTEM.
  • 37 Heaven Large collection of facts and links about this number.



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