Solve in integers the equation $x^2 + y^2 - 1 = 4xy$

Let $x$ be a positive real number, and $\lfloor{x}\rfloor$ be the largest integer that not exceeding $x$. Prove that there exist infinity number of positive integers, $n$, such that $\lfloor{\sqrt{2}}\ n\rfloor$ is a perfect square.

Show that there are infinitely many integers $n$ such that $2n + 1$ and $3n + 1$ are perfect squares, and that such $n$ must be multiples of $40$.

Show that the equation $x^2 + y^3 = z^4$ has infinitely many integer solutions.

Find all $n\in\mathbb{N}$ such that $$\binom{n}{k-1} = 2 \binom{n}{k} + \binom{n}{k+1}$$

for some natural number $k < n$.

Prove that if $m=2+2\sqrt{28n^2 +1}$ is an integer for some $n\in\mathbb{N}$, then $m$ is a perfect square.

Prove that if the difference of two consecutive cubes is $n^2$, $n\in\mathbb{N}$, then $(2n-1)$ is a square.

If $n$ is an integer such that the values of $(3n+1)$ and $(4n+1)$ are both squares, prove that $n$ is a multiple of $56$.

Let $p$ be a prime. Prove that the equation $x^2-py^2 = -1$ has integral solution if and only if $p=2$ or $p\equiv 1\pmod{4}$.

If $p$ is a prime of the form $4k+3$, show that exactly one of the equations $x^2-py^2=\pm 2$ has an integral solution.

Solve in nonnegative integers the equation $$2^x -1 = xy$$

Show that $x^4 + y^4 = z^2$ is not solvable in positive integers.

Find all primes $p$ for which there exist positive integers $x$, $y$, and $n$ such that $$p^n = x^3+y^3$$