27 (twenty-seven; Roman numeral XXVII) is the natural number following 26 and preceding 28.

Mathematics

Twenty-seven is the cube of 3, or the 2nd tetration of 3:. It is divisible by the number of prime numbers below it (9|nine). The first non-trivial decagonal number is 27. 27 has an aliquot sum of 13 (the sixth prime number) in the aliquot sequence of only one composite number, rooted in the 13-aliquot tree. In the Collatz conjecture (i.e. the 3n+1 problem), a starting value of 27 requires 3 × 37 = 111 steps to reach 1, more than any smaller number. 27 is also the fourth perfect totient number — as are all powers of 3 — with its adjacent members 15 and **3**9 adding to twice 27. A prime reciprocal magic square based on multiples of in a 6 \times 6 square has a magic constant of 27. Including the null-motif, there are 27 distinct hypergraph motifs. There are exactly twenty-seven straight lines on a smooth cubic surface, which give a basis of the fundamental representation of Lie algebra. The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional; its automorphism group is the 52-dimensional exceptional Lie algebra There are twenty-seven sporadic groups, if the non-strict group of Lie type \mathrm {T} (with an irreducible representation that is twice that of in 104 dimensions) is included. In Robin's theorem for the Riemann hypothesis, twenty-seven integers fail to hold for values where \gamma is the Euler–Mascheroni constant; this hypothesis is true if and only if this inequality holds for every larger n.

Base-specific

In decimal, 27 is the first composite number not divisible by any of its digits, as well as: Also in base ten, if one cyclically rotates the digits of a three-digit number that is a multiple of 27, the new number is also a multiple of 27. For example, 378, 783, and 837 are all divisible by 27. In senary (base six), one can readily test for divisibility by 43 (decimal 27) by seeing if the last three digits of the number match 000, 043, 130, 213, 300, 343, 430, or 513. In decimal representation, 27 is located at the twenty-eighth (and twenty-ninth) digit after the decimal point in π: If one starts counting with zero, 27 is the second self-locating string after 6, of only a few known.

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Twenty-seven is also: