Determine-all-primes-p-for-which-the-system-of-equations-p-1-2x-2-p-2-1-2y-2-has-solution-in-integers-x-y-

Question Number 120923 by bemath last updated on 04/Nov/20

Determine all primes p for which

the system of equations

{\begin{cases} p + 1 = {2 x}^{2} \\ p^{2} + 1 = {2 y}^{2} \end{cases}; has solution in

integers x, y .

Answered by liberty last updated on 04/Nov/20

Note that p + 1 = 2 x^{2} is even so \neq 2 also

{2 x}^{2} \equiv 1 \equiv {2 y}^{2} (\mod p) which implies x = \pm y . Since p is odd

and x < y < p we have x + y = p, then

p^{2} + 1 = 2 {(p - x)}^{2} = 2 p^{2} - 4 p x + p + 1

g i v e p = 4 x - 1; {2 x}^{2} = 4 x, x is 0 or 2 .

and p is - 1 or 7 . Of course - 1 is not

prim \bar{e}, but p = 7 is prime and (x, y) = (2, 5) is a solution

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