NL-NWO 2014 Problem 4

A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if

• $p$ is an odd prime number,
• $a, b$ , and $c$ are distinct and
• $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$ . a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ . b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3}$