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37 (thirty-seven) is the natural number following 36 and preceding 38.
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
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).
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:
- 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]
- The atomic number of rubidium
- The normal human body temperature in degrees Celsius
Astronomy
- NGC 2169 is known as the 37 Cluster, due to its resemblance of the numerals.
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
- List of highways numbered 37
- Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania
- I37 (disambiguation)
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