Czechoslovak Mathematical Journal, Vol. 71, No. 3, pp. 689-696, 2021
On the Diophantine equation $(2^x-1)(p^y-1)=2z^2$
Ruizhou Tong
Received February 11, 2020. Published online March 6, 2021.
Abstract: Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2\nmid x$, $p\equiv\pm3\pmod8,$ the Diophantine equation $(2^x-1)(p^y-1)=2z^2$ has no positive integer solution except when $p=3$ or $p$ is of the form $p=2a_0^2+1$, where $a_0>1$ is an odd positive integer. (2) if $2\nmid x$, $2\mid y$, $y\neq2,4,$ then the Diophantine equation $(2^x-1)(p^y-1)=2z^2$ has no positive integer solution.
References: [1] Z. Cao: On the Diophantine equation $x^{2n}-Dy^2=1$. Proc. Am. Math. Soc. 98 (1986), 11-16. DOI 10.1090/S0002-9939-1986-0848864-4 | MR 0848864 | Zbl 0596.10016
[2] Z. Cao: Introduction to Diophantine Equations. Harbin Institute of Technology Press, Harbin (1989). (In Chinese.) MR 1029025 | Zbl 0849.11029
[3] J. H. E. Cohn: The Diophantine equation $(a^n-1)(b^n-1)=x^2$. Period. Math. Hung. 44 (2002), 169-175. DOI 10.1023/A:1019688312555 | MR 1918683 | Zbl 1012.11024
[4] X. Guo: A note on the Diophantine equation $(a^n-1)(b^n-1)=x^2$. Period. Math. Hung. 66 (2013), 87-93. DOI 10.1007/s10998-012-6964-8 | MR 3018202 | Zbl 1274.11089
[5] L. Hajdu, L. Szalay: On the Diophantine equation $(2^n-1)(6^n-1)=x^2$ and $(a^n-1)(a^{kn}-1)=x^2$. Period. Math. Hung. 40 (2000), 141-145. DOI 10.1023/A:1010335509489 | MR 1805312 | Zbl 0973.11015
[6] G. He: A note on the exponential Diophantine equation $(a^m-1)(b^n-1)=x^2$. Pure Appl. Math. 27 (2011), 581-585. (In Chinese.) MR 2906439 | Zbl 1249.11056
[7] R. Keskin: A note on the exponential Diophantine equation $(a^n-1)(b^n-1)=x^2$. Proc. Indian Acad. Sci., Math. Sci. 129 (2019), Article ID 69, 12 pages. DOI 10.1007/s12044-019-0520-x | MR 3993857 | Zbl 1422.11071
[8] J. Luo: On the Diophantine equation $\frac{a^nx^m\pm1}{a^nx\pm1}=y^n+1$. J. Sichuan Univ., Nat. Sci. Ed. 36 (1999), 1022-1026. (In Chinese.) MR 1746962 | Zbl 0948.11016
[9] A. Noubissie, A. Togbé: A note on the exponential Diophantine equation $(a^n-1)(b^n-1)=x^2$. Ann. Math. Inform. 50 (2019), 159-165. DOI 10.33039/ami.2019.11.002 | MR 4048812 | Zbl 07174847
[10] L. Szalay: On the Diophantine equation $(2^n-1)(3^n-1)=x^2$. Publ. Math. 57 (2000), 1-9. MR 1771666 | Zbl 0961.11013
[11] M. Tang: A note on the exponential Diophantine equation $(a^m-1)(b^n-1)=x^2$. J. Math. Res. Expo. 6 (2011), 1064-1066. DOI 10.3770/j.issn:1000-341X.2011.06.014 | MR 2896318 | Zbl 1265.11065
[12] R. W. van der Waall: On the Diophantine equations $x^2+x+1=3v^2$, $x^3-1=2y^2$, $x^3+1=2y^2$. Simon Stevin 46 (1972), 39-51. MR 0316374 | Zbl 0246.10011
[13] P. G. Walsh: On Diophantine equations of the form $(x^n-1)(y^m-1)=z^2$. Tatra Mt. Math. Publ. 20 (2000), 87-89. MR 1845448 | Zbl 0992.11029
[14] P. Yuan, Z. Zhang: On the Diophantine equation $(a^n-1)(b^n-1)=x^2$. Publ. Math. 80 (2012), 327-331. DOI 10.5486/PMD.2012.5004 | MR 2943006 | Zbl 1263.11045
Affiliations: Ruizhou Tong, Editorial Department of Journal, Chaoyang Teachers College, No. 966, Section 4, Longshan Street, Shuangta District, Chaoyang, Liaoning 122000, P. R. China, e-mail: 806481866@qq.com
