Abstract:
This article discusses the existence of the perfect powers obtained by the result of
the addition and subtraction of two Fibonacci numbers, namely Fn and Fm under
the condition n ≡ m (mod 2). In the case of n > m > 0, the addition or subtraction
of Fn and Fm gives a product of FN and LM with N = (n±m)/2 and M = (n±m)/2
under the condition of n ≡ m (mod 4) or n ≡ m + 2 (mod 4). Then the product
of FN and LM is a perfect power in the form of 2syp with s ≥ 0, y ≥ 1, and p ≥ 2
and it must satisfy the solution (N,M) of the product of FN and LM, for N ≤ 24
and M ≤ 12.