On the equation F(n 2 + 1) = bF(n 2 − 1) + c

Under a suitable conjecture concerning prime numbers, we provide a complete solution to the equation F(n 2 + 1) = bF(n 2 − 1) + c for every n ∈ N, n > 1, where F : N → C is a completely multiplicative function and b, c ∈ C, c ̸= 0.