Step-by-step explanation:
where p is a positive prime number.
2. Preliminaries
The Catalan’s conjecture is a well known conjecture. This conjecture states
that (3, 2, 2, 3) is a unique solution (a, b, x, y) for the Diophantine equation
a
x − b
y = 1 where a, b, x and y are integers with min{a, b, x, y} > 1. In 2004,
this conjecture was proven in 2004 by Mihailescu [3].
Proposition 2.1. [3] (3, 2, 2, 3) is a unique solution (a, b, x, y) for
the Diophantine equation a
x − b
y = 1 where a, b, x and y are integers with
min{a, b, x, y} > 1.
Next, we will prove two Lemmas by Proposition 2.1.
Lemma 2.2. (1, 3) is a unique solution (x, z) for the Diophantine equation
8
x + 1 = z
2 where x and z are non-negative integers.
Proof. Let x, y and z be non-negative integers such that 8x + 1 = z
2
. If
x = 0, then z
2 = 2 which is impossible. Then x ≥ 1. Thus, z
2 = 8x + 1 ≥
8
1 + 1 = 9. Then z ≥ 3. Now, we consider on the equation z
2 − 8
x = 1. By
Proposition 2.1, we have x = 1. Then z = 3. Hence, (1, 3) is a unique solution
(x, z) for the equation 8x + 1 = z
2 where x and z are non-negative integers.
Lemma 2.3. The Diophantine equation 1 + 19y = z
2 has no non-negative
integer solution.
Proof. Suppose that there are non-negative integers y and z such that 1 +
19y = z
2
. If y = 0, then z
2 = 2 which is impossible. Then y ≥ 1. Thus,
z
2 = 1 + 19y ≥ 1 + 191 = 20. Then z ≥ 5. Now, we consider on the equation
z
2 − 19y = 1. By Proposition 2.1, we have y = 1. Then z
2 = 20. This is a
contradiction. Hence, the equation 1 + 19y = z
2 has no non-negative integer
solution.
3. Results
In [4], the Diophantine equation 8x + 19y = z
2 has no non-negative integer
solution. But we will show in this section that (1, 0, 3) is a unique solution
(x, y, z) for the Diophantine equation 8x + 19y = z
2 where x, y and z are non
