## 16) Consider the equation: gx – hx = j g = h, j = 0 and g, h, and j are non- negative integers, then the equation has how

Question

16) Consider the equation: gx – hx = j
g = h, j = 0 and g, h, and j are non-
negative integers, then the equation has
how many solutions?

in progress 0
3 years 2021-07-23T15:35:25+00:00 1 Answers 16 views 0

1. 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