Problems & Puzzles: Puzzles

 Puzzle 461. Consecutive squares + prime = prime. JM Bergot sent the following nice puzzle: The first seven even squares are 4, 16, 36, 64, 100, 144, 196. To each I can add the prime 67 to sum to a prime. Can you find a prime like 67 to which you can add more than the first seven even squares to get a prime? Q. Find a better solution changing or the set of consecutive even squares or the prime, or both.

Contributions came from Enoch Haga, Torbjörn Alm, Jan van Delden, J. K. Andersen, Farideh Firoozbakht & W Edwin Clark.

All of them found that:

163 is the only prime p up to 10^10 such that p+(2n)^2 for n = 1, 2, 3, ..., 19 are prime.

Farideh wrote: 163 is the only number p up to 3*10^10 such that p+4m^2 for m=0,1,2,...,19  are prime.

But only Andersen experimented with other series of consecutive even squares and found that:

Some long prime runs which do not start with the first even squares:
n^2+1333963 for 19 even n from 1090 to 1126.
n^2+177616147 for 19 even n from 9194 to 9230.
n^2+94180717 for 22 even n from 41806 to 41848.

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