(untitled) (European Journal of Pure and Applied Mathematics, 2026)
This mathematical research investigates whether the sum of two powers of Mersenne numbers can be expressed as a square. It also determines whether the sum of powers of these special numbers can be a power of two. Diophantine analysis using elementary methods and well-established results in number theory served as the basis for the work. The results show that there are infinitely many Mersenne numbers of the form 22α −1 whose sum of powers can be written as a perfect square and that the sum of the zeroth power and the first power of Mersenne numbers can be written as powers of 2. Moreover, the sum of two zeroth powers of Mersenne numbers always yields 2. Similarly, the sum of any two positive powers of 1, the first positive Mersenne number, is always equal to 2.