31

This number is a prime.

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The earliest and the only known case such that the sum of the divisors of two distinct numbers (16 and 25) is the same prime quantity (31), that is to say: 1+2+4+8+16 = 31; 1+5+25 = 31. [Rivera]

The smallest prime that starts a run of exactly four distinct primes that remain prime after each digit d is replaced by d^3: 31, 271, 83431, 5122764271. [Rivera]

31#/2 +/- 2 are the fourth and largest known (as of 2020) "cousin primes" of the type p#/2 +/- 2. (100280245063 and 100280245067). The other three pairs occur for p = 5, 7 and 13. [Rivera]