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On some Diophantine equations and applications Goenka, Ritesh


The Lebesgue-Nagell equation, x² + D = yⁿ, n ≥ 3 integer, is a classical family of Diophantine equations that has been extensively studied for decades. Bugeaud, Mignotte, and Siksek [20] made a landmark contribution to its study by resolving the equation for all values of D satisfying 1 ≤ D ≤ 100. However, many cases where -100 ≤ D ≤ -1 remain an open problem. We build on the works of Carlos [6] and Chen [22] to establish new results in several cases from this regime. Our techniques involve a combination of linear forms in logarithms and the modular method applied to Q-curves. Bennett et al. [10] proved using the theory of Diophantine equations that the Fourier coefficients of the modular discriminant form, or equivalently, values of the Ramanujan tau function, are never equal to the power of an odd prime smaller than 100. We generalize these inadmissibility results to other Hecke newforms with rational integer Fourier coefficients and trivial mod 2 residual Galois representation. In doing so, we use some results about the Lebsgue-Nagell equation and several techniques, including the modular method, Thue-Mahler equations, and the Primitive Divisor theorem for Lucas sequences.

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