What are the whole numbers for which two times the forth power of it plus one is a square?
In mathematical notation: $$2n^4+1=m^2;m,n\in\mathbb{Z}$$
My Observations:
- because of the squares, all negative numbers can be exchanged for their additive inverse and still give a solution (the problem reduces to the natural numbers containing 0)
- $2n^4=(m+1)(m-1)$: each of the same prime factors of $n^4$ (except 2) have to divide completely into either $m+1$ or $m-1$ because not both can be divisible by a prime (bigger than 2), e.g. only $m+1$ is divisible by $3^8$ and $3∤m-1$
- modular arithmetic does not seem to get one very far