IJPAM: Volume 104, No. 4 (2015)

THE COMPLETE SET OF SOLUTIONS OF
THE DIOPHANTINE EQUATION $p^x+q^y=z^2$
FOR TWIN PRIMES $p$ AND $q$

Jerico B. Bacani$^1$, Julius Fergy T. Rabago$^2$
$^{1,2}$Department of Mathematics and Computer Science
College of Science
University of the Philippines Baguio
Governor Pack Road, Baguio City 2600, PHILIPPINES


Abstract. 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

\begin{displaymath} p^x+q^y=z^2 \end{displaymath} (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 $(p,q,x,y,z)\in\{(17,19,1,1,6),(71,73,1,1$, $12)\}$ also satisfies equation ([*]) (cf. [4]).

In the present paper, we give more solutions to ([*]). That is, we show that if the well-known Twin Prime Conjecture is true, then the Diophantine equation given by ([*]), 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 ([*]) has the unique solution $(x,y,z)=(1,1,\sqrt{p+q})$ in non-negative integers.

Received: May 17, 2015

AMS Subject Classification: 11D61

Key Words and Phrases: Diophantine equation, twin primes, integer solutions

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DOI: 10.12732/ijpam.v104i4.3 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 104
Issue: 4
Pages: 517 - 521


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