On perfect powers that are sums of two Pell numbers

On perfect powers that are sum of two balancing numbers

“…Further, in [19] Zhang and Togbé studied the Diophantine equations F q n ± F q m = y p (1.6) with positive integers q, p ≥ 2 and gcd(F n , F m ) = 1. Also, perfect powers that are sums of two Pell numbers has been studied (see [1]). Recently, in [4] Bhoi et al, study the Diophantine equation U n + U m = x q in integers n ≥ m ≥ 0, x ≥ 2, and q ≥ 2, where (U k ) ≥0k is Lucas sequence of first kind.…”

“…Let d = gcd(n, m). Then(1) gcd(B n , B m ) = B d , (2) gcd(C n , C m ) = C d if a = b and is 1 otherwise, (3) gcd(B n , C m ) = C d if a > band is 1 otherwise. Lemma 2.8.…”

A note on the exponential Diophantine equation (A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z

“…has yielded very rich results, but some important problems about it are far from being solved (see [7]). In recent 20 years, many authors have considered equation (1) when A, B and C are Fibonacci A00045 or Lucas A000204 or Pell A000129 numbers in OEIS [14] or when A and B are Fibonacci or Lucas or Pell numbers (see [1,2,4,6,12,13,18]).…”