On a Diophantine Equation Involving Lucas Numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910025189949440 |
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| author | Ibrahimov, Seyran S. Mahmudov, Nazim I. |
| author_facet | Ibrahimov, Seyran S. Mahmudov, Nazim I. |
| contents | Let L_t denote the t-th Lucas number. We prove that the Diophantine equation
L_m^{n+k} + L_m^n = L_r
has no solutions in positive integers r, m, n, and k with m >= 2. In the case n = 1, the proof is based on a precise factorization formula for the difference of two Lucas numbers and the Carmichael Primitive Divisor Theorem. For n >= 2, we apply lower bounds for linear forms in logarithms due to Matveev, combined with Legendre's lemma, an exact divisibility property for powers of Lucas numbers, and computer-assisted computations to complete the proof. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_14379 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Diophantine Equation Involving Lucas Numbers Ibrahimov, Seyran S. Mahmudov, Nazim I. Number Theory 11B39, 11D61, 11B83 Let L_t denote the t-th Lucas number. We prove that the Diophantine equation L_m^{n+k} + L_m^n = L_r has no solutions in positive integers r, m, n, and k with m >= 2. In the case n = 1, the proof is based on a precise factorization formula for the difference of two Lucas numbers and the Carmichael Primitive Divisor Theorem. For n >= 2, we apply lower bounds for linear forms in logarithms due to Matveev, combined with Legendre's lemma, an exact divisibility property for powers of Lucas numbers, and computer-assisted computations to complete the proof. |
| title | On a Diophantine Equation Involving Lucas Numbers |
| topic | Number Theory 11B39, 11D61, 11B83 |
| url | https://arxiv.org/abs/2506.14379 |