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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 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 ''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 \tfrac 1e\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 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 Heaven
Large collection of facts and links about this number. {{Integers, zero Integers