Problems & Puzzles: Puzzles

 Puzzle 514. S(pi)= q mod r Farideh Firoozbakht poses the following nice puzzle: 2 + 3 = 5 (mod 7) 2 + 3 + 5 + ... + 31 = 37 (mod 41) 2 + 3 + 5 + ... + 6563 = 6569 (mod 6571) Note that 6563, 6569 & 6571 are consecutive primes. What is the next term of the sequence 7, 41, 6571, ... ?

No solutions were reported by Jacques Tramu, Torbjörn Alm, J. K. Andersen, Luca Poletti, Giovanni Resta & Farid Lian.

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All of them reported no more solutions. The largest upper limit of the search was 55,167,866,355,403 by Resta.

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It appears the chance of getting a particular value modulo p would be 1/p.
The sum of the reciprocals of the primes diverges slowly, so I expect
infinitely many solutions but far apart.
The expected number of solutions between x and y is the
integral for n = x to y of 1/(n*log n). 1/log n is the chance that n is prime.
That gives 0.50 expected solutions from 5*10^12 to 10^21, and 0.98
from 5*10^12 to 10^34.

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