# Lemon and melon

**Number Theory**Level 4

In a two-dimensional plane there are some vectors. We call a vector as a *lemon vector* if it has positive integer coordinates, and we call a vector a *melon vector* if its coordinates are primes.

We know that \(\overrightarrow { u }\), \(\overrightarrow{ v }\) and \(\overrightarrow { w }\) are three lemon vectors, and \(\overrightarrow { u+v }\), \(\overrightarrow { v+w }\) and \(\overrightarrow{ w+u }\) are three melon vectors. How many different \(\overrightarrow { v }\) vectors there are if \(\overrightarrow { u }\), \(\overrightarrow { v }\) and \(\overrightarrow { w }\) have the same angle but different module?