Given 3n algebraic integers α_i,ν, i=1...n, ν=0,1,2, and an integer ideal q in an algebraic number field K, we obtain several new bounds on the number of solutions to the congruence with a quadratic exponential polynomial
Σ_i=1ⁿΠ_ν²α_i,ν^xν≡ 0 (mod q), 1 ≤ x ≤ N. We then apply these bounds to studying arithmetic properties of values of linear recurrence sequences on squares.

I.S. was supported in part by the ARC Grant DP180100201.