- Thirty-one is the third 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.
- 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.
- 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).
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