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May 20, 2015 at 4:05 comment added Max Alekseyev @GerryMyerson: Oh, I did not notice. Anyway, no search bound was specified explicitly so far, but here it comes.
May 20, 2015 at 4:01 comment added The Masked Avenger Also, many of the examples can be sieved out with small primes, for example all those candidates for p that are 2 mod 7.
May 20, 2015 at 3:57 comment added Gerry Myerson This counterexample was in Linus Hamilton's comment on knsam's (not-a-complete-)answer.
May 20, 2015 at 3:21 comment added Max Alekseyev @Turbo: Solutions to the Pellian equation (explained above by knsam) grow exponentially fast, so it is not a big deal to test them all below a bound like $10^{5000}$. In fact, I tested smallest $10^4$ solutions and achieved the bound $10^{5722}$.
May 20, 2015 at 3:16 comment added Turbo How did you get that number?
May 20, 2015 at 3:11 comment added Max Alekseyev @Turbo: Yes. What is your concern?
May 20, 2015 at 3:08 comment added Turbo $10^{5\times 10^3}$?
May 20, 2015 at 2:36 history edited Max Alekseyev CC BY-SA 3.0
bound updated
May 20, 2015 at 2:19 history edited Max Alekseyev CC BY-SA 3.0
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May 20, 2015 at 2:07 history answered Max Alekseyev CC BY-SA 3.0