Ain't nothing new it's getting old.
Now you're standing in the cold.
Play that same old song and dance.
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You done had it baby.
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31 is the 3rd Mersenne prime ( 25 - 1 ) as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496, since 496 = 25 - 1 ( 25 - 1). 31 is the eighth Mersenne prime exponent. 31 is also the 4th lucky prime and the 11th supersingular prime.
31 is a centered triangular number, a centered pentagonal number and centered decagonal number. At 31, the Mertens function sets a new low of -4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31, making 31 a self number.
31 is a repdigit in base 5 (111), and base 2 (11111).
The numbers 31, 331, 3331, 33331, 333331, 3333331, and 33333331 are all prime. For a time it was thought that every number of the form 3w1 would be prime. However, the next nine numbers of the sequence are composite; their factorisations are:
333 333 331 = 17*19607843
3 333 333 331 = 673*4952947
33 333 333 331 = 307*108577633
333 333 333 331 = 19*83*211371803
3 333 333 333 331 = 523*3049*2090353
33 333 333 333 331 = 607*1511*1997*18199
333 333 333 333 331 = 181*1841620626151
3 333 333 333 333 331 = 199*16750418760469
33 333 333 333 333 331 = 31*1499*717324094199.
The recurrence of the factor 31 in the last number above can be used to prove that no sequence of the type RwE or ERw can consist only of primes because every prime in the sequence will periodically divide further numbers. Here, 31 divides every fifteenth number in 3w1 (and 331 every 110th).
27 is a perfect cube, being 3³ = 3 × 3 × 3. . 27 is 23 (see tetration). There are exactly 27 straight lines on a smooth cubic surface, which give a basis of the fundamental representation of the E6 Lie algebra. 27 is also a decagonal number.
27 has an aliquot sum of 13 and is the first composite member of the 13-aliquot tree with the aliquot sequence (27,13,1,0). Twenty-seven is the aliquot sum of the two odd discrete semiprimes 69 and 133.
In base 10, it is the first composite number not evenly divisible by any of its digits. It is the radix (base) of the septemvigesimal positional numeral system.
In a prime reciprocal magic square of the multiples of 1/7, the magic constant is 27.
In the Collatz conjecture (aka the "3n + 1 conjecture") a starting value of 27 requires 112 steps to reach 1, many more than any lower number.
The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional.[1]
In base 10, it is a Smith number and a Harshad number.
It is the twenty-eighth (and twenty-ninth) digit in π. (3.141592653589793238462643383279...).
It is the atomic number of cobalt.
Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. Twenty-three is also the fifth factorial prime, the third Woodall prime. It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 23 is a number.
The fifth Sophie Germain prime and the fourth safe prime, 23 is the next to last member of the first Cunningham chain of the first kind to have five terms (2, 5, 11, 23, 47). Since 14! + 1 is a multiple of 23 but 23 is not one more than a multiple 14, 23 is a Pillai prime. 23 is the smallest odd prime to be a highly cototient number, as the solution to x − φ(x) for the integers 95, 119, 143, 529.
Twenty-three is the aliquot sum of two integers; the discrete semiprimes 57 and 85 and is the base of the 23-aliquot tree.
23 is the first prime P for which unique factorization of cyclotomic integers based on the Pth root of unity breaks down.
The sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers.
In the list of fortunate numbers, 23 occurs twice, since adding 23 to either the fifth or eighth primorial gives a prime number (namely 2333 and 9699713).
23 also has the distinction of being one of two integers that cannot be expressed as the sum of fewer than 9 cubes of integers (the other is 239). See Waring's problem.
23 is a Wedderburn–Etherington number. The codewords in the perfect (non-extended) binary Golay code are of size 23.
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