×

zbMATH — the first resource for mathematics

On a diophantine equation. (English) Zbl 0905.11018
The equation \(x^2+ 3^{2k} =y^n\) where \(n\geq 3\) is studied. For \(n=3\), it is proved that it has a solution only if \(k\equiv 2\pmod 3\) and then it is unique. If this equation has a solution for \(n>3\), then \(n\) is odd and \(k=2^\delta \kappa\), where \(\delta\geq 1\), \((2,\delta) =1\), \(\kappa\equiv 15\pmod{20}\) and all the prime divisors \(p\) of \(n\) are congruent to \(11 \pmod {12}\).
Reviewer: E.L.Cohen (Ottawa)

MSC:
11D41 Higher degree equations; Fermat’s equation
11D61 Exponential Diophantine equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1007/BF02589348 · Zbl 0064.04007 · doi:10.1007/BF02589348
[2] Ljunggren, Kong. Norsk. Vid. Selskab Forh. Trond. 16 pp 27– (1943)
[3] Lebesgue, Von. Ann. Des. Math 9 pp 178– (1850)
[4] DOI: 10.1007/BF01197049 · Zbl 0770.11019 · doi:10.1007/BF01197049
[5] Störmer, Bull. Soc. Math. 27 pp 160– (1899)
[6] DOI: 10.1155/S0161171297000409 · Zbl 0881.11038 · doi:10.1155/S0161171297000409
[7] Cohn, Acta Arith. 65 pp 367– (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.