On a diophantine equation. (English) Zbl 0905.11018

Abu Muriefah, Fadwa S.; Arif, S. Akhtar

Bull. Aust. Math. Soc. 57, No. 2, 189-198 (1998).

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

Keywords:

higher degree diophantine equations; exponential equations

Citations:

Zbl 0905.11017

References:

[1]	DOI: 10.1007/BF02589348 · Zbl 0064.04007
[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
[5]	Störmer, Bull. Soc. Math. 27 pp 160– (1899)
[6]	DOI: 10.1155/S0161171297000409 · Zbl 0881.11038
[7]	Cohn, Acta Arith. 65 pp 367– (1993)