Solve the equation with natural numbers \(1 + x + x^2 + x^3 = 2y\).
Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).
Prove there are no integer solutions for the equation \(x^2 + 1990 = y^2\).
Let \(p\) and \(q\) be two prime numbers such that \(q = p + 2\). Prove that \(p^q + q^p\) is divisible by \(p + q\).
Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).