For such equations:
Using the solutions of the Pell equation. $p^2-(t^4-4)s^2=1$
You can write the solution.
It all comes down to the Pell equation - as I said.
Considering specifically the equation:
Decisions are determined such consistency. Where the next value is determined using the previous one.
You start with numbers. $(p_1;s_1) - (55 ; 12)$
Using these numbers, the solution can be written according to a formula.
If you use an initial $(p_1 ; s_1) - (1 ; 1)$
Then the solutions are and are determined by formula.
As the sequence it is possible to write endlessly. Then the solutions of the equation, too, can be infinite.
If you use a sequence with the first element. $ (p ; s ) $ - $( 4 ; 1 )$
Then decisions can be recorded.
If you use a sequence with the first element. $( p ; s )$ - $( 55 ; 12 )$
Using this sequence can be different. On its basis with the first element.
$( z ; q )$ - $(2 ; 1 )$
Then decisions will be.
It is necessary to take into account that the number can have a different sign. - $(p ; s )$