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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
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##### References:
 [1] S. K. Jena, Method of infinite ascent applied on mA6 + nB3 = C2, Math. Student, Vol. 77 (2008), 239-246. [2] S. K. Jena, Method of infinite ascent applied on A4±nB2 = C3, Math. Student, Vol. 78 (2009), 233-238. · Zbl 1221.11084 [3] S. K. Jena, Method of infinite ascent applied on mA3 + nB3 = C2, Math. Student, Vol. 79 (2010), 187-192. [4] S. K. Jena, Beyond the method of infinite descent, J. Comb. Inf. Syst. Sci., Vol 35 (2010), 501-511. [5] S. K. Jena, Method of infinite ascent applied on mA3 + nB3 = 3C2, Math. Student, Vol. 81 (2012), 151-160. [6] 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 [7] 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 [8] H. Cohen, Number Theory-Volume I: Tools and Diophantine Equations, Springer-Verlag, GTM 239, 2007. · Zbl 1119.11001 [9] H. Cohen, Number Theory-Volume II: Analytic and Modern Tools, Springer-Verlag, GTM 240, 2007. · Zbl 1119.11002 [10] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, GTM 106, 1986; Expanded 2nd ed., 2009. · Zbl 0585.14026
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