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The Diophantine equation \((2am - 1)^x + (2m)^y = (2am + 1)^z\). (English) Zbl 1290.11068

Recently, A. Tobgé and T. Miyazaki [Glasg. Math. J. 51, No. 3, 659–667 (2009; Zbl 1194.11044)] gave all the solutions of the Diophantine equation \[ n^x+(n+1)^y=(n+2)^z\tag{1} \] in positive integers. In this paper, the authors extend the equation (1) by giving all of the solutions of the equation \[ (2am-1)^x+(2m)^y=(2am+1)^z\tag{2} \] where \(m,x,y,z\in\mathbb{Z}^{+}\) and \(a>1\) and \(a\) is odd as \((m,x,y,z)=(2a,2,2,2),(1,1,1,1),\) further, \((m,x,y,z)=(\sqrt{a},2,3,2)\), when \(a\) is a square and the additional solution \((m,x,y,z)=(1,1,13,2)\) when \(a=45\). Further, using this main result, they prove that \[ b^x+2^y=(b+2)^z\tag{3} \] has only the solution \((x,y,z)=(1,1,1)\), if \(b\neq 89\) and the solutions \((x,y,z)=(1,1,1)\), \((1,13,2)\) if \(b=89\) where \(b\) is an odd positive integer and \(b\geq 5\).
Roughly speaking, their proof goes as follows. First, the authors consider the case \(y>1\). When \(m>1\), they use only elementary methods. In the case \(m=1\), they use W. Ivorra’s result [Acta Arith. 108, No.4, 327–338 (2003; Zbl 1026.11035)] (see Proposition 2.1) which is based on the theory of modular forms and Galois representations and also the result of M. Bauer and M. Bennett which is based on Diophantine approximations. Secondly, they consider the case \(y=1\). They complete the proof of the main result by means of M. Laurent’s result [Acta Arith. 133, No. 4, 325–348 (2008; Zbl 1215.11074)] (see proposition 2.3) which uses an application of lower bounds for linear forms in logarithms. Finally, using M. H. Le’s result [Arch. Math. 78, No. 1, 26–35 (2002; Zbl 1006.11013)] (see Proposition 3.1) and the main result of their paper recalled in (2) they find all the solutions of the equation (3).

MSC:

11D61 Exponential Diophantine equations
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[1] DOI: 10.1023/A:1015779301077 · Zbl 1010.11020 · doi:10.1023/A:1015779301077
[2] DOI: 10.1007/BF01197049 · Zbl 0770.11019 · doi:10.1007/BF01197049
[3] Dem’janenko V. A., Izv. Vyssh. Ucebn. Zaved. Mat. 173 pp 29–
[4] DOI: 10.1017/S0017089509990073 · Zbl 1194.11044 · doi:10.1017/S0017089509990073
[5] DOI: 10.4064/aa108-4-3 · Zbl 1026.11035 · doi:10.4064/aa108-4-3
[6] DOI: 10.4064/aa133-4-3 · Zbl 1215.11074 · doi:10.4064/aa133-4-3
[7] DOI: 10.1007/s00013-002-8213-5 · Zbl 1006.11013 · doi:10.1007/s00013-002-8213-5
[8] Terai N., Acta Arith. 90 pp 17–
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