Equation $x^2=y^p+1$

Question

can you help me please for solving this dophantine equation

$$x^2=y^p+1$$

and if you can give me a general method to studying such equation

$$x²=y^p+t$$

Thanks

Answer 1

Partial answer

In general $x^2-1=y^p$. For $x$ even, $x-1,x+1$ are odd and relatively prime, and this it would mean $x-1=y_0^p$ and $x+1=y_1^p$. This is impossible unless $x=0$ and $p$ odd.

If $y$ is even, then $x$ is odd and $x-1$ and $x+1$ have only a factor of $2$ in common.

Thus, one of them must be of the form $2^{p-1}y_0^p$ and the other of the form $2y_1^p$ and $y_1$ odd. So you need to find $y_0$ and $y_1$ so that $$\left|2^{p-2}y_0^p-y_1^p\right|=1$$

This is definitely not possible of $p$ even, except when $y_1=1$ and $y_0=0$, since the left side is the difference of squares in that case.