^{1}

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^{1}

In this paper, we study two Diophantine equations of the type
p
^{x} + 9
^{y} =
z
^{2} , where
p is a prime number. We find that the equation 2
^{x} + 9
^{y} =
z
^{2} has exactly two solutions (
x,
y,
z) in non-negative integer
i.e., {(3, 0, 3),(4, 1, 5)} but 5
^{x} + 9
^{y} =
z
^{2} has no non-negative integer solution.

Recently, there have been a lot of studies about the Diophantine equation of the type

In this study, we consider the Diophantine equation of the type

Theorem 2.1. (Catalan’s Conjecture [

Theorem 2.2. The Diophantine equation

Proof: Let x and z be non-negative integers such that

Case-1: If

Case-2: If

Hence,

Theorem 2.3. The Diophantine equation

Proof: Let x and z be non-negative integers such that

or

Let

or

Thus,

Therefore,

Corollary 2.4. The Diophantine equation

Theorem 2.5. The Diophantine equation

Proof: Suppose x and z be non-negative integers such that

or

Thus,

Theorem 2.6. The Diophantine equation

Proof: Suppose

Now we consider the following remaining cases.

Case-1:

and

Case-2:

Case-3:

Case-4:

Let

The Diophantine Equation (1) is a Diophantine equation by Catalan’s type

Theorem 2.7. The Diophantine equation

Proof: Suppose

Now we consider the following remaining cases.

Case-1:

Case-2:

Case-3:

Case-4:

Let

In the paper, we have discussed two Diophantine equation of the type

Md. Al-Amin Khan,Abdur Rashid,Md. Sharif Uddin, (2016) Non-Negative Integer Solutions of Two Diophantine Equations 2^{x} + 9^{y} = z^{2} and 5^{x} + 9^{y} = z^{2}. Journal of Applied Mathematics and Physics,04,762-765. doi: 10.4236/jamp.2016.44086