About a recurrence equation of prime numbers

Question

Let $p$ be a prime.

Consider the recurrence equation $$s_{n}=(s_{n-1}²-2)(mod(2^{p}-1))$$ where $s₀=4$

My question is: Can we write this recurrence as follow? $$s_{n}=(2^{p}-1)q+(s_{n-1}²-2)$$ where $q$ is an integer.

Answer 1

This follow by definition of congruency if $x\equiv b(mod\ c)$ then$x=b+c\cdot q$ where q is an integer.Substitute b=$s_{n-1}^2-2$ ,$c=2^p-1$,x=$s_{n}$ .