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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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[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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