Article
| Title: | A remark on a Diophantine equation of S. S. Pillai (English) |
| Author: | Hoque, Azizul |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 3 |
| Year: | 2024 |
| Pages: | 897-903 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | S. S. Pillai proved that for a fixed positive integer $a$, the exponential Diophantine equation $x^y-y^x= a$, $\min (x,y)>1$, has only finitely many solutions in integers $x$ and $y$. We prove that when $a$ is of the form $2z^2$, the above equation has no solution in integers $x$ and $y$ with $\gcd (x,y)=1$. (English) |
| Keyword: | Pillai's Diophantine equation |
| Keyword: | Lehmer sequence |
| Keyword: | primitive divisor |
| MSC: | 11D61 |
| MSC: | 11D72 |
| DOI: | 10.21136/CMJ.2024.0124-24 |
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| Date available: | 2024-10-03T12:39:49Z |
| Last updated: | 2024-10-04 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152587 |
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| Reference: | [9] Pillai, S. S.: On the indeterminate equation $x^y-y^x = a$.Annamalai Univ. J. 1 (1932), 59-61. Zbl 0005.05302 |
| Reference: | [10] Robbins, N.: Fibonacci numbers of the form $cx^2$, where $1 \leq c \leq 1000$.Fibonacci Q. 28 (1990), 306-315. Zbl 0728.11013, MR 1077496 |
| Reference: | [11] Voutier, P. M.: Primitive divisors of Lucas and Lehmer sequences.Math. Comput. 64 (1995), 869-888. Zbl 0832.11009, MR 1284673, 10.1090/S0025-5718-1995-1284673-6 |
| Reference: | [12] Waldschmidt, M.: Perfect powers: Pillai's works and their developments.Collected works of S. Sivasankaranarayana Pillai. Volume 1 Ramanujan Mathematical Society, Mysore (2010), xxii--xlvii. MR 2766491 |
| Reference: | [13] Yuan, P.: On the Diophantine equation $ax^2 + by^2 = ck^n$.Indag. Math., New Ser. 16 (2005), 301-320. Zbl 1088.11024, MR 2319301, 10.1016/S0019-3577(05)80030-8 |
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