Let p be a prime number. We construct a directed graph of p vertices, labeled with integers from 0 to p−1. There is an edge from vertex x to vertex y if and only if x2+1≡y(modp). Let f(p) denotes the length of the longest directed cycle in this graph. Prove that f(p) can attain arbitrarily large values. number theoryprime numbersQuadratic ResiduesDirected graphsquadratics