Find all integer values of $$x$$ such that $$x^2+13x+3$$ is a perfect integer square.

Question

Uniteasy Admin · Accepted Answer

Let $$m^2=x^2+13x+3$$
$$4m^2=4x^2+4	imes 13x+13^2-13^2+12 $$
$$(2x+13)^2-(2m)^2+169-12=157)$$
$$(2x+13-2m)(2x+13+2m) =157$$
Let $$a=2x+13-2m$$, $$b=2x+13+2m$$
$$	herefore  x=\dfrac{a+b-26}{4} $$
And
$$a\cdotp b=157=1	imes157  $$
Therefore 
$$a=1,b=157$$
$$x=33$$
$$m^2 = 33^2+13	imes 33+3 = 1521$$
$$ m = 39$$

Question

Answer