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September 23, 2025Open Access

Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations

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Key Points

All integer solutions to the equation have been classified, revealing unique structures not previously identified.
The equation under study incorporates components that link powers of primes, notably featuring bases like 3, 19, and 73.
By integrating the primitive divisor theorem, the research contributes pivotal understanding to the behavior of similar equations.
Findings validate that the role of algebraic integer rings is essential in framing solutions within a broader mathematical context.

Abstract

In this paper, we provide a complete classification of the integer solutions to the Diophantine equation x² + 3ᵃ 19ᵇ 73ᶜ = yⁿ, where \ (= 2^ \), \ (x, y 1 \), \ (a, b, c, 0 \), \ (n 3 \), \ ( (x, y) = 1 \). Our approach combines the Primitive Divisor Theorem for Lehmer sequences, proved by Bilu, Hanrot, and Voutier, with fundamental properties of algebraic integer rings. By employing these methods, we determine all possible solutions in non-negative integers.

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Authors

Murat Alan

Yıldız Technical University

M. Evren Aydin

Yıldız Technical University

Journals

Communications in Advanced Mathematical Sciences

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Institutions

Yıldız Technical University

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Cite this study

Alan et al. (Tue,) studied this question.

synapsesocial.com/papers/68d4759931b076d99fa6d9f9 — DOI: https://doi.org/10.33434/cams.1675491

Also consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

On the solutions of some Lebesgue–Ramanujan–Nagell type equations· 2024 · 2 citations
On the Diophantine equation $$x^2+C=y^n$$· 2022 · 3 citations
On the Diophantine equation $$x^2+3^a41^b=y^n $$· 2020 · 7 citations
The Magma Algebra System I: The User Language· 1997 · 7,392 citations
ON THE EXPONENTIAL DIOPHANTINE EQUATIONx2+p2m=2yn· 2012 · 9 citations

Also consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

On the solutions of some Lebesgue–Ramanujan–Nagell type equations· 2024 · 2 citations
On the Diophantine equation $$x^2+C=y^n$$· 2022 · 3 citations
On the Diophantine equation $$x^2+3^a41^b=y^n $$· 2020 · 7 citations
The Magma Algebra System I: The User Language· 1997 · 7,392 citations
ON THE EXPONENTIAL DIOPHANTINE EQUATIONx2+p2m=2yn· 2012 · 9 citations

Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations

Key Points

Abstract

Citation Network

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Discussion

Authors

Journals

Actions

Institutions

References and Citations

Citation Network

Connected Papers

Discussion

Cite this study

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Also consider