For an odd number \(N\) denote by \(A\) the minimal positive difference between prime divisors of \(N\), denote by \(B\) the minimal positive difference between composite divisors of \(N\). Usually we have \(A>B\), but can we have \(B>A\)?
Let \(n\) be an integer bigger than \(1\), and \(p\) a prime number. Suppose that \(n\) divides \(p-1\) and \(p\) divides \(n^3-1\). Prove that \(4p-3\) is a square number.
Let \(n\) be a natural number, not \(1\) and not prime. Arrange the factors of \(n\) greater than \(1\) in a circle. When can this be done such that neighbours in the circle are never coprime?
Let \(x\), \(y\), \(z\) and \(w\) be non-negative integers. Find all solutions to \(2^x3^y-5^z7^w=1\).