31:
31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge.
31 is the third Mersenne prime (25 − 1)[1] and the eighth Mersenne prime exponent, 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 = 2(5 − 1)(25 − 1). 31 is also the 4th lucky prime[2] and the 11th supersingular prime.[3]
31 is a centered triangular number,[4] the lowest prime centered pentagonal number[5] and a centered decagonal number.[6]
For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.
At 31, the Mertens function sets a new low of −4, a value which is not subceded until 110.
31 is a centered pentagonal number
No integer added up to its base 10 digits results in 31, making 31 a self number.[7]
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
