37 (number)

← 36 37 38 →
← 30 31 32 33 34 35 36 37 38 39 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
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

37 is the 12th prime number, and the 3rd isolated prime without a twin prime.

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

37 is the first irregular prime with irregularity index of 1, where the smallest prime number with an irregularity index of 2 is the thirty-seventh prime number, 157.

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

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).

37 requires twenty-one steps to return to 1 in the 3x + 1 Collatz problem, as do adjacent numbers 36 and 38. The two closest numbers to cycle through the elementary {16, 8, 4, 2, 1} Collatz pathway are 5 and 32, whose sum is 37; 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, 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.

In moonshine theory, whereas all p ⩾ 73 are non-supersingular primes, the smallest such prime is 37.

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

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. 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 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.

In science

• 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.

In other fields

House number in Baarle (in its Belgian part)

Thirty-seven is:

• The number of the French department Indre-et-Loire
• 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"
• More often than expected, the number given by people when asked to provide a random 2 digit number.

See also

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

External links

• 37 Heaven Large collection of facts and links about this number.