Concept
32 (thirty-two) is the natural number following 31 and preceding 33. 32 is the fifth power of two (), making it the first non-unitary fifth-power of the form p5 where p is prime. 32 is the totient summatory function over the first 10 integers, and the smallest number with exactly 7 solutions for . The aliquot sum of a power of two () is always one less than the number itself, therefore the aliquot sum of 32 is 31. The product between neighbor numbers of 23, the dual permutation of the digits of 32 in decimal, is equal to the sum of the first 32 integers: . 32 is the ninth 10-happy number, while 23 is the sixth. Their sum is 55, which is the tenth triangular number, while their difference is . 32 is also a Leyland number expressible in the form , where: On the other hand, a regular 32-sided icosidodecagon contains distinct symmetries. There are collectively 32 uniform colorings to the 11 regular and semiregular tilings. The product of the five known Fermat primes is equal to the number of sides of the largest regular constructible polygon with a straightedge and compass that has an odd number of sides, with a total of sides numbering The first 32 rows of Pascal's triangle in binary represent the thirty-two divisors that belong to this number, which is also the number of sides of all odd-sided constructible polygons with simple tools alone (if the monogon is also included). There are 32 three-dimensional crystallographic point groups and 32 five-dimensional crystal families, and the maximum determinant in a 7 by 7 matrix of only zeroes and ones is 32. In sixteen dimensions, the sedenions generate a non-commutative loop of order 32, and in thirty-two dimensions, there are at least 1,160,000,000 even unimodular lattices (of determinants 1 or −1); which is a marked increase from the twenty-four such Niemeier lattices that exists in twenty-four dimensions, or the single lattice in eight dimensions (these lattices only exist for dimensions ).