# zbMATH — the first resource for mathematics

The method of infinite ascent applied on $$2^p A^6 + B^3 = C^2$$. (English) Zbl 1306.11026
Summary: In this paper, we prove that for any positive integer $$p$$, when $$p \equiv 1\pmod 6$$ or, $$p \equiv 3\pmod 6$$, the Diophantine equation: $$2^p A^6 +B^3 = C^2$$ has infinitely many co-prime integral solutions $$A, B, C$$. When $$p = 0$$, this equation has only four integral solutions with $$(A, B, C) = (\pm 1, 2, \pm 3)$$. For other integer values of $$p$$, the problem is open.
##### MSC:
 11D41 Higher degree equations; Fermat’s equation 11D72 Diophantine equations in many variables
Full Text:
##### References:
  S. K. Jena, Method of infinite ascent applied on mA6 + nB3 = C2, Math. Student, Vol. 77 (2008), 239-246.  S. K. Jena, Method of infinite ascent applied on A4±nB2 = C3, Math. Student, Vol. 78 (2009), 233-238. · Zbl 1221.11084  S. K. Jena, Method of infinite ascent applied on mA3 + nB3 = C2, Math. Student, Vol. 79 (2010), 187-192.  S. K. Jena, Beyond the method of infinite descent, J. Comb. Inf. Syst. Sci., Vol 35 (2010), 501-511.  S. K. Jena, Method of infinite ascent applied on mA3 + nB3 = 3C2, Math. Student, Vol. 81 (2012), 151-160.  S. K. Jena, The method of infinite ascent applied on A4 ±nB3 = C2, Czech. Math. J. 63 (138) (2013), No. 2, 369-374. · Zbl 1289.11033  S. K. Jena, Method of infinite ascent applied on A3 ± nB2 = C3, Notes on Number Theory and Discrete Mathematics, Vol. 19 (2013), No. 2, 233-238. · Zbl 1221.11084  H. Cohen, Number Theory-Volume I: Tools and Diophantine Equations, Springer-Verlag, GTM 239, 2007. · Zbl 1119.11001  H. Cohen, Number Theory-Volume II: Analytic and Modern Tools, Springer-Verlag, GTM 240, 2007. · Zbl 1119.11002  J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, GTM 106, 1986; Expanded 2nd ed., 2009. · Zbl 0585.14026
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.