Abstract
Let p be an odd prime. By using a lower bound for linear forms in logarithms of two algebraic numbers, we prove that if \(p>10^{24}\), 2 is a primitive root module p and the least solution \((u_1,\ v_1)\) of Pell’s equation \(u^2-2(p-1)(p-2)v^2=1\) satisfies \(\log \left( u_1+v_1\sqrt{2(p-1)(p-2)}\right) <p^{\frac{2}{3}}\), then the equation \(\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}\) has no positive integer solutions \((x,\ y,\ m,\ n)\) with \(x=2,\ y=p\) and \(m>n>2\).
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The authors would like to thank the anonymous referees for their valuable suggestions.
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This work is supported by N.S.F. (11226038, 11371012) of P. R. China, the N.S.F. (2017JM1025) of Shaanxi Province, the Education Department Foundation of Shaanxi Province (17JK0323)
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Yang, H., Fu, R. A note on the Goormaghtigh equation. Period Math Hung 79, 86–93 (2019). https://doi.org/10.1007/s10998-018-0265-9
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DOI: https://doi.org/10.1007/s10998-018-0265-9