International Journal of Pure and Applied Mathematics Volume 104 No. 4 2015, 517-521 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v104i4.3 P A ijpam.eu 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 p x + q y = z 2 (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) ∈{(17, 19, 1, 1, 6), (71, 73, 1, 1, 12)} also satisfies equation (1) (cf. [4]). In the present paper, we give more solutions to (1). That is, we show that if the well-known Twin Prime Conjecture is true, then the Diophantine equation given by (1), 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 (1) has the unique solution (x,y,z) = (1, 1, √ p + q) in non-negative integers. AMS Subject Classification: 11D61 Key Words: Diophantine equation, twin primes, integer solutions Received: May 17, 2015 c  2015 Academic Publications, Ltd. url: www.acadpubl.eu § Correspondence author