Journal of the Ramanujan Mathematical Society
Volume 40, Issue 3, September 2025 pp. 257–263.
Solution of certain Diophantine equations
in Gaussian integers
Authors:
Arkabrata Ghosh
Author institution:SRM University-AP, Mangalgiri Mandal, Neerukonda, Amravati, Andhra Pradesh~522~502.
Summary:
In this article, we show that the quartic Diophantine equations x{4}
± pqy{4} = ± z{2} and x{4} ± pq y{4}= ± iz{2} have only trivial
solutions for some primes p and q satisfying conditions p
≡ 3 (mod 8), q ≡ 1 (mod 8) and (p/q) = −1.
Here we have found the torsion of the two families of elliptic
curves to find the solutions of given Diophantine equations.
Moreover, we also calculate the rank of these two
families of elliptic curves over the Gaussian field ℚ(i).
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