According to Wikipedia:
31 is the 3rd 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. 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).
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; their factorisations are:
333 333 331 = 17*19607843
3 333 333 331 = 673*4952947
33 333 333 331 = 307*108577633
333 333 333 331 = 19*83*211371803
3 333 333 333 331 = 523*3049*2090353
33 333 333 333 331 = 607*1511*1997*18199
333 333 333 333 331 = 181*1841620626151
3 333 333 333 333 331 = 199*16750418760469
33 333 333 333 333 331 = 31*1499*717324094199.
The recurrence of the factor 31 in the last number above can be used to prove that no sequence of the type RwE or ERw can consist only of primes because every prime in the sequence will periodically divide further numbers. Here, 31 divides every fifteenth number in 3w1 (and 331 every 110th).
I didn't understand any of that.
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