Is it true that if $p^2 - q^2$ is a perfect square where $p$ and $q$ are primes $> 5000$ then it has a prime factor greater than $17$?

Note: This question was asked in MSE but did not receive an answer. Hence posting in MO.

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Is it true that if $p^2 - q^2$ is a perfect square where $p$ and $q$ are primes $> 5000$ then it has a prime factor greater than $17$?

Note: This question was asked in MSE but did not receive an answer. Hence posting in MO.

allthe prime factors exceed $17$, but the body asks whetheroneprime factor exceeds $17$. Please edit for consistency. $\endgroup$ – Gerry Myerson Dec 28 '19 at 14:38