The main purpose of this paper is to correct the result of A. Suvarnamani that was published in this journal. In particular, A. Suvarnamani showed in [6] that (p, q, x, y, z) = (3, 5, 1, 0, 2) is the “unique solution” to the
Diophantine equation
p^x + q^y = z^2, (1)
where p is an odd prime, q−p = 2 and x, y and z are non-negative integers. The author, however, did not realize that (p, q, x, y,
... [Show full abstract] z) ∈ {(17, 19, 1, 1, 6), (71, 73, 1, 1,12)} also satisfies equation (1) (cf. [4]). In the present paper, we give more solutions to (1). That is, we show that if
the well-known Twin Prime Conjecture is true, then the Diophantine equation given by (1), where p and q are twin primes, has infinitely many solutions (p, q, x, y, z) in positive integers. Furthermore, we show that if the sum of p and q is a square, then (1) has the unique solution (x, y, z) = (1, 1,√p + q) in non-negative integers.