37 (number)

37 (thirty-seven) is the natural number following 36 and preceding 38.

← 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, xxxvii
Binary	1001012
Ternary	11013
Senary	1016
Octal	458
Duodecimal	3112
Hexadecimal	2516

In mathematics

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 ${\displaystyle 0}$ 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 ${\displaystyle {\tfrac {1}{e}}\approx 37\%}$.

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.^[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:

• 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]

Science

The 37 Cluster

• NGC 2169 is known as the 37 Cluster, due to its resemblance of the numerals.
• The average body temperature of a human in Celsius is 37 degrees.