<s>[INST] Find all prime solutions of equation $5x^2-7x+1=y^2.$ Find all prime solutions of the equation $5x^2-7x+1=y^2.$
It is easy to see that 
$y^2+2x^2=1 \mod 7.$ Since $\mod 7$-residues are $1,2,4$ it follows that $y^2=4 \mod 7$, $x^2=2 \mod 7$  or $y=2,5 \mod 7$ and $x=3,4 \mod 7.$ 
In the same way from $y^2+2x=1 \mod 5$ we have that $y^2=1 \mod 5$ and $x=0 \mod 5$ or $y^2=-1 \mod 5$ and $x=4 \mod 5.$
How put together the two cases?
Computer find two prime solutions $(3,5)$ and $(11,23).$
 [/INST] If there is a solution it has more than 4000 digits, which makes me think there is no solution beyond the one two already mentioned.
In Mathematica,
i=1;
ans=Solve[5 x ^2-7x+1==y^2&&x>0&&y>0,{x,y},Integers];
ans2=ans/.C[1]->i//RootReduce
Dynamic@i
Dynamic[IntegerLength@ans2[[All,All,2]]]

and then run
    While[FreeQ[PrimeQ[ans2[[All,All,2]]],{True,True}],i++;ans2=ans/.C[1]->i//RootReduce]

Setting i=0 will stop the loop with the two known solutions:
{{x->3,y->5},{x->11,y->23}}

but there appear to be no easily found solutions beyond i=1
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