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System of equations involving squares and primes

Source: 2022 Bulgarian Spring Math Competition, Problem 9.3

March 27, 2022

algebrasystem of equationsprime

Problem Statement

Find all primes

p

, such that there exist positive integers

x

y

which satisfy

\begin{cases} p + 49 = 2x^2\\ p^2 + 49 = 2y^2\\ \end{cases}