Multiplying two expressions containing perfect squares to get another perfect square

Question

Is it possible to multiply a perfect square by the previous square plus one and get another perfect square?

An example that doesn't work: $$6^2 (5^2 + 1) = 936 \ne n^2$$

Answer 1

The ratio of two perfect squares is a perfect square, so we have to find a perfect square in the form $(x-1)^2+1$.

Answer 2

You are basically asking for integer solutions to $y^2=x^2((x-1)^2+1)$.
$n^2$ and $n^2+1$ are both perfect squares iff $n=0$, so we can assume that $(x-1)^2+1$ isn't a perfect square (you can quickly check the case $x=1$ by hand, it is the only solution).

When you factorize $(x-1)^2+1$ it must be divisible by a prime with an odd exponent since it isn't a square, while the factorization of $x^2$ contains only primes with an even exponent so their product $x^2(x-1)^2+1$ must contain a prime with an odd exponent and hence can't be a square