Generalized Fruit Diophantine equation and Hyperelliptic curves

2023/01/31 by Prakash, Om, Chakraborty, Kalyan · 1 citation
#11D72. Secondary: 11G30 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11D41

paper · doi:10.48550/arxiv.2301.13474

Abstract

We show the insolvability of the Diophantine equation ax^d-y²-z²+xyz-b=0 in ℤ for fixed a and b such that a≡ 1 \pmod 12 and b=2^da-3, where d is an odd integer and is a multiple of 3. Further, we investigate the more general family with b=2^da-3^r, where r is a positive odd integer. As a consequence, we found an infinite family of hyperelliptic curves with trivial torsion over ℚ. We conclude by providing some numerical evidence corroborating the main results.

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• Generalized fruit Diophantine equation and super elliptic curves

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