32 is the fifth power of two (), making it the first non-unitary fifth-power of the form where is prime. 32 is the totient summatory function over the first 10 integers,[1] 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.[2]
The first 32 rows of Pascal's triangle read as single binary numbers represent the 32 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).[10]
In the Standard Model of particle physics, there are 32 degrees of freedom among the leptons and all bosons that interact with them (including the graviton, which is generally expected to exist, and assuming there are no right-handed neutrinos)[citation needed]
In the Kabbalah, there are 32 Kabbalistic Paths of Wisdom. This is, in turn, derived from the 32 times of the Hebrew names for God, Elohim appears in the first chapter of Genesis.
In chess, the total number of black squares on the board, the total number of white squares, and the total number of pieces (black and white) at the beginning of the game.
^Specifically, 31 is the eleventh prime number, equal to the sum of 20 and its composite index 11, where 33 is the twenty-first composite number, equal to the sum of 21 and its composite index 12 (which are palindromic numbers).[8][9]32 is the only number to lie between two adjacent numbers whose values can be directly evaluated from sums of associated prime and composite indices (32 is the twentieth composite number, which maps to 31 through its prime index of 11, and 33 by a factor of 11, that is the composite index of 20; the aliquot part of 32 is 31 as well).[2] This is due to the fact that the ratio of composites to primes increases very rapidly, by the prime number theorem.
^29 is the only earlier point, where there are twenty non primes, and ten primes. 40 — twice the composite index of 32 — lies between the 8th pair of sexy primes (37, 43),[18] which represent the only two points in the set of natural numbers where the ratio of prime numbers to composite numbers (up to) is 1/2. Where 68 is the forty-eighth composite, 48 is the thirty second, with the difference 68 – 48 = 20, the composite index of 32.[8] Otherwise, thirty-two lies midway between primes (23, 41), (17, 47) and (3, 61). At 33, there are 11 numbers that are prime and 22 that are not, when considering instead the set of natural numbers that does not include 0. The product 11 × 33 = 363 represents the thirty-second number to return 0 for the Mertens functionM(n).[19]
^Sloane, N. J. A. (ed.). "Sequence A000225 (a(n) equal to 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-08.
^Conway, John H.; Guy, Richard K. (1996). "The Primacy of Primes". The Book of Numbers. New York, NY: Copernicus (Springer). pp. 137–142. doi:10.1007/978-1-4612-4072-3. ISBN 978-1-4612-8488-8. OCLC 32854557. S2CID 115239655.
^Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.9 Archimedean and uniform colorings". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 102–107. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. S2CID 119730123.
^Cawagas, Raoul E.; Gutierrez, Sheree Ann G. (2005). "The Subloop Structure of the Cayley-Dickson Sedenion Loop" (PDF). Matimyás Matematika. 28 (1–3). Diliman, Q.C.: The Mathematical Society of the Philippines: 13–15. ISSN 0115-6926. Zbl 1155.20315.
^Baez, John C. (November 15, 2014). "Integral Octonions (Part 8)". John Baez's Stuff. U.C. Riverside, Department of Mathematics. Retrieved 2023-05-04.