In this study, we will use Catalan’

In this study, we will use Catalan’s conjecture (see [2]), which states that the
only solution in integers a > 1, b > 1, x > 1, y > 1 to the equation a
x − b
y = 1
is (a, b, x, y) = (3, 2, 2, 3).
Now we will consider the Diophantine equation p
x + (p + 1) = z
2
, where p
is a Mersenne prime.
Theorem 1. The Diophantine equation p
x + (p + 1)y = z
2
, where p
is a Mersenne prime, has only two solutions in non-negative integer, namely,
(p, x, y, z) = (7, 0, 2, 3) and (p, x, y, z) = (3, 2, 2, 5).
Proof. Let p be a Mersenne prime. Then p = 2q − 1 for some prime q. It is
easy to check that z must be odd and p ≡ 3(mod 4). Thus z
2 ≡ 1(mod 4) and
p + 1 ≡ 0(mod 4).
Now we consider the Diophantine equation p
x+(p+1)y = z
2
in the following
cases:
Case 1 : Suppose x = 0. It can be checked easily that, if the Diophantine
equation 1 + (p + 1)y = z
2 has a solution, then