How to find the smallest $p,q\in\mathbb N$ such that the following expression gives a periodic sequence of $a$ ODD numbers and $b$ EVEN numbers? ($n=0,1,2,\dots$) $$ S(n)=p^n\bmod q$$
Some examples below, written as "$(a,b) : p \space q \space\space\space\space s$", where $s(n)=S(n)\bmod2$
(1,1) : 2 3 [1, 0]
(1,2) : 2 7 [1, 0, 0]
(1,3) : 2 15 [1, 0, 0, 0]
(1,4) : 2 31 [1, 0, 0, 0, 0]
(1,5) : 2 63 [1, 0, 0, 0, 0, 0]
(2,1) : 7 9 [1, 1, 0]
(2,2) : 3 5 [1, 1, 0, 0]
(2,3) : 37 41 [1, 1, 0, 0, 0]
(2,4) : 11 21 [1, 1, 0, 0, 0, 0]
(2,5) : 89 111 [1, 1, 0, 0, 0, 0, 0]
(3,1) : 3 19 [1, 1, 1, 0]
(3,2) : 11 25 [1, 1, 1, 0, 0]
(3,3) : 5 9 [1, 1, 1, 0, 0, 0]
(3,4) : 45 71 [1, 1, 1, 0, 0, 0, 0]
(3,5) : 53 153 [1, 1, 1, 0, 0, 0, 0, 0]
(4,1) : 3 11 [1, 1, 1, 1, 0]
(4,2) : 9 35 [1, 1, 1, 1, 0, 0]
(4,3) : 15 49 [1, 1, 1, 1, 0, 0, 0]
(4,4) : 9 17 [1, 1, 1, 1, 0, 0, 0, 0]
(4,5) : 9 259 [1, 1, 1, 1, 0, 0, 0, 0, 0]
(5,1) : 31 57 [1, 1, 1, 1, 1, 0]
(5,2) : 11 43 [1, 1, 1, 1, 1, 0, 0]
(5,3) : 165 203 [1, 1, 1, 1, 1, 0, 0, 0]
(5,4) : 19 81 [1, 1, 1, 1, 1, 0, 0, 0, 0]
(5,5) : 19 25 [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]
The examples were searched for by brute force.
How can one find the solutions directly, or in the most efficient way?
Will there always be solutions for every $a,b$ ?
The only pattern I know of so far is the most trivial one: $(1,b) : p,q=2,2^{b+1} - 1$