Problems & Puzzles:
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
the prime, or
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
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.