37 (number)

37 is the 12th prime number, and the 3rd isolated prime without a twin prime.{{Cite OEIS |A007510 |Single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime. |access-date=2022-12-05 }}

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|37 is the third star number{{Cite web|url=https://oeis.org/A003154|title=Sloane's A003154: Centered 12-gonal numbers. Also star numbers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-31}} and the fourth centered hexagonal number.{{Cite web|url=https://oeis.org/A003215|title=Sloane's A003215: Hex (or centered hexagonal) numbers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-31}}

|The sum of the squares of the first 37 primes is divisible by 37.{{cite OEIS|A111441|Numbers k such that the sum of the squares of the first k primes is divisible by k|access-date=2022-06-02}}

|37 is the median value for the second prime factor of an integer.{{Cite book |last=Koninck |first=Jean-Marie de |title=Those fascinating numbers |last2=Koninck |first2=Jean-Marie de |date=2009 |publisher=American Mathematical Society |isbn=978-0-8218-4807-4 |location=Providence, R.I}}

|Every positive integer is the sum of at most 37 fifth powers (see Waring's problem).{{Cite web|last=Weisstein|first=Eric W.|title=Waring's Problem|url=https://mathworld.wolfram.com/WaringsProblem.html|access-date=2020-08-21|website=mathworld.wolfram.com|language=en}}

|It is the third cuban prime following 7 and 19.{{Cite web|url=https://oeis.org/A002407|title=Sloane's A002407: Cuban primes|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-31}}

|37 is the fifth Padovan prime, after the first four prime numbers 2, 3, 5, and 7.{{Cite web|url=https://oeis.org/A000931|title=Sloane's A000931: Padovan sequence|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-31}}

|It is the fifth lucky prime, after 3, 7, 13, and 31.{{Cite web|url=https://oeis.org/A031157|title=Sloane's A031157: Numbers that are both lucky and prime|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-31}}

|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.

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37 is the first irregular prime with irregularity index of 1,{{Cite web|url=https://oeis.org/A000928|title=Sloane's A000928: Irregular primes|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-31}} where the smallest prime number with an irregularity index of 2 is the thirty-seventh prime number, 157.{{Cite OEIS |A073277 |Irregular primes with irregularity index two. |access-date=2024-03-25 }}

The smallest magic square, using only primes and 1, contains 37 as the value of its central cell:{{Cite book |author=Henry E. Dudeney |author-link=Henry Dudeney |title=Amusements in Mathematics |publisher=Thomas Nelson & Sons, Ltd. |location=London |year=1917 |page=125 |url=http://djm.cc/library/Amusements_in_Mathematics_Dudeney_edited02.pdf |archive-url=https://web.archive.org/web/20230201074043/http://djm.cc/library/Amusements_in_Mathematics_Dudeney_edited02.pdf |archive-date=2023-02-01 |url-status=live |isbn=978-1153585316 |oclc=645667320 }}

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).{{Cite web|url=https://oeis.org/A040017|title=Sloane's A040017: Unique period primes|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-31}}

37 requires twenty-one steps to return to 1 in the {{math|3x + 1}} Collatz problem, as do adjacent numbers 36 and 38.{{Cite OEIS |A006577 |Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached. |access-date=2023-09-18 }} The two closest numbers to cycle through the elementary {16, 8, 4, 2, 1} Collatz pathway are 5 and 32, whose sum is 37;{{Cite web |last=Sloane |first= N. J. A. |author-link=Neil Sloane |url=http://oeis.org/wiki/3x%2B1_problem |title=3x+1 problem |website=The On-Line Encyclopedia of Integer Sequences |publisher=The OEIS Foundation |access-date=2023-09-18 }} also, the trajectories for 3 and 21 both require seven steps to reach 1. On the other hand, the first two integers that return $0$ for the Mertens function (2 and 39) have a difference of 37,{{Cite OEIS |A028442 |Numbers k such that Mertens's function M(k) (A002321) is zero. |access-date=2023-09-02 }} 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.{{Cite OEIS |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. |access-date=2023-09-02 }}

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.{{Cite OEIS |A013629 |Floor of imaginary parts of nontrivial zeros of Riemann zeta function. }} It is in equivalence with the sum of ceilings of the first two such zeroes, 15 and 22.{{Cite OEIS |A092783 |Ceiling of imaginary parts of nontrivial zeros of Riemann zeta function. }}

The secretary problem is also known as the 37% rule by $\backslash tfrac\; 1e\backslash approx\; 37\backslash \%$.

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.{{Cite journal|last=Vukosav|first=Milica|date=2012-03-13|title=NEKA SVOJSTVA BROJA 37|url=https://hrcak.srce.hr/81042|journal=Matka: Časopis za Mlade Matematičare|language=hr|volume=20|issue=79|pages=164|issn=1330-1047}} 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.

There are precisely 37 complex reflection groups.

In three-dimensional space, the most uniform solids are:

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|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).

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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.{{Cite journal |last=Har'El |first=Zvi |url=http://harel.org.il/zvi/docs/uniform.pdf |title=Uniform Solution for Uniform Polyhedra |journal=Geometriae Dedicata |volume=47 |pages=57–110 |publisher=Springer Publishing |location=Netherlands |year=1993 |doi=10.1007/BF01263494 |mr=1230107 |zbl=0784.51020 |s2cid=120995279 }}
See, 2. THE FUNDAMENTAL SYSTEM.

File:NGC_2169.jpg

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

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• [http://thirty-seven.org 37 Heaven] Large collection of facts and links about this number.

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Category:Integers