Problems & Puzzles: Puzzles

Puzzle 505. P(m^2 + 1) = n^2 + 1 JM Bergot sent the following puzzle: P(5^2 + 1) = P(26) = 101 = 10^2 +1 P(2015^2 + 1) = P(4,060,226) = 68,956,417= 8304^2 + 1 Can you find more primes of the form P(m^2 + 1) = n^2 + 1?

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Contributions came from Farideh Firoozbakht & J. K. Andersen

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Farideh wrote:

There is no other prime of the form P(m^2 + 1) = n^2 + 1 up to m = 1133000
(except P(0^2 + 1) = 2 = 1^2 +1) .

Also I didn't find prime of the form P(m^2^k + 1) = n^2^k + 1 for m>0 and k>1 .

Note that if k has odd prime factor and m>0 then there is no prime of the form
P(m^k + 1) = n^k + 1 .

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Andersen wrote:

There are no other solutions with n < 2,000,000.

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