For what primenumbers, $p$, is $\sqrt{5p +49}$ an integer? For what primenumbers, $p$, is $\sqrt{5p +49}$ an integer?
I managed to figure out:
$5p +49 = n^2$
$5p = (n+7)(n-7)$
But can't think of anything more than that, anyone able to solve this without bruteforcing?
For primenumbers under $10^7$, the solutions are $3$ and $19$.

Then $n-7=5$ and $n+7=p$, that is $n=12$ and $p=19$.
Or $n-7=-5$ and $n+7=-p$, that is $n=2$ and $p=-9$ which is negative.
Or $n+7=5$ and $n-7=p$, that is $n=-2$ and $p=-9$ which is negative.
Or $n+7=-5$ and $n-7=-p$, that is $n=-12$ and $p=-19$, which is negative.
Or $n-7=1$ and $n+7=5p$, that is $n=8$ and $p=3$.
Or $n-7=-1$ and $n+7 = -5p$, that is $n=6$ and $p=-13/5$ which is not an integer.
Or $n+7=1$ and $n-7=5p$, that is $n=-6$ and $p=-13/5$ which is not an integer.
Or $n+7=-1$ and $n-7=5p$, that is $n=-8$ and $p=-3$ which is negative.
So in conclusion, the only possible numbers are $p= 3, 19$.
