37 (thirty-seven) is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following
36 and preceding
38.
In mathematics
37 is the 12th
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, and the 3rd isolated prime without a
twin prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin 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
In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...
, using only
primes and
1, contains 37 as the value of its central
cell:
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
Collatz problem, as do adjacent numbers
36 and
38.
The two closest numbers to cycle through the elementary 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
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s that return
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 ''k''
2 − ''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.
37 is the sixth
floor
A floor is the bottom surface of a room or vehicle. Floors vary from wikt:hovel, simple dirt in a cave to many layered surfaces made with modern technology. Floors may be stone, wood, bamboo, metal or any other material that can support the ex ...
of
imaginary parts of non-trivial
zeroes in the
Riemann zeta function.
It is in equivalence with the sum of ceilings of the first two such zeroes,
15 and
22.
The
secretary problem is also known as the ''37% rule'' by
.
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
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
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...
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
A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
solids are:
In total, these number twenty-one figures, which when including their
dual polytopes (i.e. an extra
tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
, 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.
See, 2. THE FUNDAMENTAL SYSTEM.
Astronomy

*
NGC 2169 is known as the 37 Cluster, due to its resemblance of the numerals.
References
External links
37 HeavenLarge collection of facts and links about this number.
{{Integers, zero
Integers