Section alpha_nat_coprime.
Let A_eq_3_12 n : exists u v, u*α (S n) + v*α n = 1.
End alpha_nat_coprime.
End Pell_inner.

Theorem find_odd_alpha u : exists n, (u <= alpha_nat (S n) /\ rem (alpha_nat (S n)) 2 = 1)%nat.