37 (number)
| ||||
---|---|---|---|---|
Cardinal | thirty-seven | |||
Ordinal | 37th (thirty-seventh) | |||
Factorization | prime | |||
Prime | 12th | |||
Divisors | 1, 37 | |||
Greek numeral | ΛΖ´ | |||
Roman numeral | XXXVII | |||
Binary | 1001012 | |||
Ternary | 11013 | |||
Senary | 1016 | |||
Octal | 458 | |||
Duodecimal | 3112 | |||
Hexadecimal | 2516 |
37 (thirty-seven) is the natural number following 36 and preceding 38.
In mathematics
[edit]37 is the 12th prime number, and the 3rd isolated prime without a twin prime.[1]
- 37 is the third star number[2] and the fourth centered hexagonal number.[3]
- The sum of the squares of the first 37 primes is divisible by 37.[4]
- 37 is the median value for the second prime factor of an integer.[5]
- Every positive integer is the sum of at most 37 fifth powers (see Waring's problem).[6]
- It is the third cuban prime following 7 and 19.[7]
- 37 is the fifth Padovan prime, after the first four prime numbers 2, 3, 5, and 7.[8]
- It is the fifth lucky prime, after 3, 7, 13, and 31.[9]
- 37 is a sexy prime, being 6 more than 31, and 6 less than 43.
- 37 remains prime when its digits are reversed, thus it is also a permutable prime.
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 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 . 37 is also known to be the most common answer when asked for a number between 1 and 100, thus an example of the blue–seven phenomenon, and being humanly pseudo-random
Decimal properties
[edit]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.
Also the midpoint of 73 is 37, being half of 74. The digits not in common, 3 and 4, add up to the other digit 7; while 3 × 4 = 12, the prime index.
Geometric properties
[edit]There are precisely 37 complex reflection groups.
In three-dimensional space, the most uniform solids are:
- the five Platonic solids (with one type of regular face)
- the fifteen Archimedean solids (counting enantimorphs, all with multiple regular faces); and
- the sphere (with only a singular facet).
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]
In science
[edit]- The atomic number of rubidium
- The normal human body temperature in degrees Celsius
Astronomy
[edit]- NGC 2169 is known as the 37 Cluster, due to its resemblance of the numerals.
In other fields
[edit]Thirty-seven is:
- The number of the French department Indre-et-Loire[22]
- 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"
- "Evo Moment 37", widely considered the most iconic moment in the history of competitive video gaming
See also
[edit]- List of highways numbered 37
- Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania
- I37 (disambiguation)
References
[edit]- ^ 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.
- ^ "Sloane's A003154: Centered 12-gonal numbers. Also star numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
- ^ "Sloane's A003215: Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
- ^ 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.
- ^ Koninck, Jean-Marie de; Koninck, Jean-Marie de (2009). Those fascinating numbers. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-4807-4.
- ^ Weisstein, Eric W. "Waring's Problem". mathworld.wolfram.com. Retrieved 2020-08-21.
- ^ "Sloane's A002407: Cuban primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
- ^ "Sloane's A000931: Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
- ^ "Sloane's A031157: Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
- ^ "Sloane's A000928: Irregular primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
- ^ 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.
- ^ 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.
- ^ "Sloane's A040017: Unique period primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
- ^ 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.
- ^ Sloane, N. J. A. "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-09-18.
- ^ 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.
- ^ 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.
- ^ Sloane, N. J. A. (ed.). "Sequence A013629 (Floor of imaginary parts of nontrivial zeros of Riemann zeta function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A092783 (Ceiling of imaginary parts of nontrivial zeros of Riemann zeta function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Vukosav, Milica (2012-03-13). "NEKA SVOJSTVA BROJA 37". Matka: Časopis za Mlade Matematičare (in Croatian). 20 (79): 164. ISSN 1330-1047.
- ^ 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. - ^ Département d'Indre-et-Loire (37), INSEE
External links
[edit]- 37 Heaven Large collection of facts and links about this number.