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method of infinite ascent; Diophantine equation $A^4 \pm nB^3 = C^2$
Each of the Diophantine equations $A^4 \pm nB^3 = C^2$ has an infinite number of integral solutions $(A, B, C)$ for any positive integer $n$. In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when $A$, $B$ and $C$ are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions $(A, B, C)$ of the Diophantine equation $aA^3 + cB^3 = C^2$ for any co-prime integer pair $(a,c)$.
[1] Beukers, F.: The Diophantine equation $Ax^p + By^q = Cz^r$. Duke Math. J. 91 (1998), 61-88. DOI 10.1215/S0012-7094-98-09105-0 | MR 1487980
[2] Jena, S. K.: Method of infinite ascent applied on $A^4 \pm nB^2 = C^3$. Math. Stud. 78 (2009), 233-238. MR 2779731
[3] Jena, S. K.: Method of infinite ascent applied on $mA^3 + nB^3 = C^2$. Math. Stud. 79 (2010), 187-192. MR 2906833
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