TSWJ The Scientific World Journal 1537-744X 2356-6140 Hindawi Publishing Corporation 10.1155/2015/306590 306590 Research Article The Diophantine Equation 8x+py=z2 Qi Lan 1 Li Xiaoxue 2 Bellouquid Abdelghani 1 College of Mathematics and Statistics Yulin University Yulin, Shaanxi 719000 China yulinu.edu.cn 2 School of Mathematics Northwest University Xi’an, Shaanxi 710127 China nwu.edu.cn 2015 1412015 2015 07 07 2014 14 12 2014 1412015 2015 Copyright © 2015 Lan Qi and Xiaoxue Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p±3(mod  8), then the equation 8x+py=z2 has no positive integer solutions (x,y,z); (ii) if p7(mod  8), then the equation has only the solutions (p,x,y,z)=(2q-1,(1/3)(q+2),2,2q+1), where q is an odd prime with q1(mod  3); (iii) if p1(mod  8) and p17, then the equation has at most two positive integer solutions (x,y,z).

1. Introduction

Let Z, N be the sets of all integers and positive integers, respectively. Let p be a fixed odd prime. Recently, the solutions (x,y,z) of the equation (1)8x+py=z2,x,y,zN were determined in the following cases:

(Sroysang ) if p=19, then (1) has no solutions;

(Sroysang ) if p=13, then (1) has no solutions;

(Rabago ) if p=17, then (1) has only the solutions (x,y,z)=(1,1,5), (2,1,9), and (3,1,23).

In this paper, using certain results of exponential Diophantine equations, we prove a general result as follows.

Theorem 1.

If p±3(mod  8), then (1) has no solutions (x,y,z). If p7(mod  8), then (1) has only the solutions (2)p,x,y,z=2q-1,q+23,2,2q+1, where q is an odd prime with q1(mod  3).

If p1(mod  8) and p17, then (1) has at most two solutions (x,y,z).

Obviously, the above theorem contains the results of [1, 2]. Finally, we propose the following conjecture.

Conjecture 2.

If p17, then (1) has at most one solution (x,y,z).

2. Preliminaries Lemma 3.

If 2n-1 is a prime, where n is a positive integer, then n must be a prime.

Proof.

See Theorem 1.10.1 of .

Lemma 4.

If p is an odd prime with p1(mod  4), then the equation (3)u2-pv2=-1,u,vN has solutions (u,v).

Proof.

See Section 8.1 of .

Lemma 5.

The equation (4)X2-2m=Yn,X,Y,m,nN,gcdX,Y=1,Y>1,m>1,n>2 has only the solution (X,Y,m,n)=(71,17,7,3).

Proof.

See Theorem 8.4 of .

Lemma 6.

Let D be a fixed odd positive integer. If the equation (5)u2-Dv2=-1,u,vN has solutions (u,v), then the equation (6)X2-D=2n,X,nN,n>2 has at most two solutions (X,n), except the following cases:

D=22r-3·2r+1+1, (X,n)=(2r-3,3), (2r-1,r+2), (2r+1,r+3), and (3·2r-1,2r+3), where r is a positive integer with r3;

D=1/3(22r+1-17)2-32, (X,n)=1/3(22r+1-17),5, 1/3(22r+1+1,2r+3), and 1/3(17·22r+1-1),4r+7, where r is a positive integer with r3;

D=22r1+22r2-2r1+r2+1-2r1+1-2r2+1+1, (X,n)=(2r2-2r1-1,r1+2), (2r2-2r1+1,r2+2), and (2r2+2r1-1,r1+r2+2), where r1, r2 are positive integers with r2>r1+1>2.

Proof.

See .

Lemma 7.

If D is an odd prime and D belongs to the exceptional case (i) of Lemma 6, then D=17.

Proof.

We now assume that D is an odd prime with D=22r-3·2r+1+1. Then we have (7)2r-12-2r+2=D,(8)2r+12-2r+3=D. If 2r, since r3, then r4, and by (7), we have (9)2r-1+2r/2+1=D,2r-1-2r/2+1=1. But, by the second equality of (9), we get 1(2r-1)-2r/2+1-1(mod  8), a contradiction.

If 2r, then from (8) we get (10)2r+1+2(r+3)/2=D,2r+1-2(r+3)/2=1. Further, by the second equality of (10), we have 2r=2(r+3)/2, r=3, and D=17. Thus, the lemma is proved.

Lemma 8.

If D is an odd prime and D belongs to the exceptional case (iii) of Lemma 6, then D=17.

Proof.

Using the same method as in the proof of Lemma 7, we can obtain this lemma without any difficulty.

Lemma 9.

If D belongs to the exceptional case (ii), then (6) has at most one solution (X,n) with 3n.

Proof.

Notice that, for any positive integer r, there exists at most one number of 5, 2r+3, and 4r+7 which is a multiple of 3. Thus, by Lemma 6, the lemma is proved.

Lemma 10.

The equation (11)Xm-Yn=1,X,Y,m,nN,minX,Y,m,n>1 has only the solution (X,Y,m,n)=(3,2,2,3).

Proof.

See .

3. Proof of Theorem

We now assume that (x,y,z) is a solution of (1). Then we have gcd(2p,z)=1.

If 2y, since gcd(z+py/2,z-py/2)=2, then from (1) we get (12)z+py/2=23x-1,z-py/2=2, where we obtain (13)z=23x-2+1,(14)py/2=23x-2-1. Since p>1, applying Lemma 10 to (14), we get (15)y=2,p=23x-2-1. Further, by Lemma 3, we see from the second equality of (15) that (16)p=2q-1,q=3x-2 is an odd prime with q1(mod3).

Therefore, by (13), (15), and (16), we obtain the solutions given in (2).

Obviously, if p satisfies (2), then p7(mod  8). Otherwise, since 2y, we see from (1) that ppyz2-8x1(mod  8). It implies that if p±3(mod  8), then (1) has no solutions (x,y,z). If p7(mod  8), then (1) has only the solutions (2).

Here and below, we consider the remaining cases that p1(mod  8). By the above analysis, we have 2y. If y>1, then y3 and (4) has the solution (X,Y,m,n)=(z,p,3x,y) with 3m. But, by Lemma 5, it is impossible. Therefore, we have (17)y=1. Substituting (17) into (1), the equation (18)X2-p=2n,X,nN,n>2 has the solution (X,n)=(z,3x) with 3n. Since p1(mod  8), by Lemma 4, (3) has solutions (u,v). Therefore, by Lemmas 69, (1) has at most two solutions (x,y,z). Thus, the theorem is proved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (11371291).

Sroysang B. More on the diophantine equation 8x + 19y = z2 International Journal of Pure and Applied Mathematics 2012 81 4 601 604 2-s2.0-84871740320 Sroysang B. On the Diophantine equation 8x+13y=z2 International Journal of Pure and Applied Mathematics 2014 90 1 69 72 10.12732/ijpam.v90i1.9 2-s2.0-84892915277 Rabago J. F. T. On an open problem by B. Sroysang Konuralp Journal of Mathematics 2013 1 2 30 32 Hua L. G. An Introduction to Number Theory 1979 Beijing, China Science Press Mordell L. J. Diophantine Equations 1969 London, UK Academic Press MR0249355 Bennett M. A. Skinner C. M. Ternary Diophantine equations via Galois representations and modular forms Canadian Journal of Mathematics 2004 56 1 23 54 10.4153/cjm-2004-002-2 MR2031121 Le M.-H. On the number of solutions of the generalized Ramanujan-Nagell equation x2-D=2n+2 Acta Arithmetica 1991 60 2 149 167 Mihăilescu P. Primary cyclotomic units and a proof of Catalan's conjecture Journal für die Reine und Angewandte Mathematik 2004 2004 572 167 195 10.1515/crll.2004.048 MR2076124