Corollary Pell_alpha_nat' b x y : 2 <= b -> x*x+y*y = 1+b*(x*y) -> { n | x = alpha_nat b n }.

Theorem alpha_nat_2lm b n m l j v : 2 <= b -> v = alpha_nat b (S (S m)) - alpha_nat b m -> arem n (S m) l j -> rem (alpha_nat b n) v = rem (alpha_nat b j) v \/ rem (alpha_nat b n + alpha_nat b j) v = 0.