This article considers the family of elliptic curves given by Ep,q:y2=x3-pqx, where p and q are distinct odd primes. More specifically, we show that p≡7(mod8), q≡5(mod8), and pq=-1, then the ranks of both Ep,q(Q) and Ep,q(Q(i)) are 0. In the second theorem, we prove that if (i) p≡3(mod8), (ii) q≡5(mod8), (iii) pq=-1, and (iv) 2(p+q) is a perfect square, then the rank of Ep,q(Q) is 1 and the rank of Ep,q(Q(i)) is 2. Finally, by using the BSD and parity conjectures, we prove that if (i) p≡5(mod8), (ii) q≡3(mod8), and (iii) pq=-1, then the rank of Ep,q(Q) is 1 over Q and the rank of Ep,q(Q(i)) is 2.

In this article, we study the family of elliptic curves given by Ep:y2=x3-5px for an odd prime p≠5 and try to find the conditions under which Ep has rank one. Specifically, we have shown that if p≡3,27(mod40) such that 2(p+5) is a perfect square or if p≡11,19(mod40) such that (5+4p) is a perfect square, then the Mordell-Weil rank of Ep is exactly one. Finally, assuming the parity conjecture, we have proved that if p≡3,11,19,27(mod40), then the rank of Ep is always one.

In this article, we demonstrate the total number of solutions of the given Diophantinc equation xyz = x + y + z = 2 over the finite fieldq = ℤ/qℤ, where q = pn with n ≥ 1 and p is a prime greater than 5. Furthermore, we quantify the total number of solutions for the given Diophantinc equation within the finite field q.

We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic twists, whereas to find the other family, we use a combination of elliptic and hyperelliptic curves. Two main results are used, one from Soleng [13] and the other from Banerjee and Hoque [3].

This article considers the family of elliptic curves given by and certain conditions on odd primes p and q. More specifically, we have shown that if and, then the rank of is zero over both and. Furthermore, if the primes p and q are of the form and, where such that is a perfect square, then the given family of elliptic curves has rank one over and rank two over. Finally, we have shown that the torsion of over is isomorphic to.

We calculate the volume of the tropical Prym variety of a harmonic double cover of metric graphs having non-trivial dilation. We show that the tropical Prym variety behaves discontinuously under deformations of the double cover that change the number of connected components of the dilation subgraph.

In this article, we consider a family of elliptic curves defined by Em: y2 = x3 − m2x + (pqr)2 where m is a positive integer and p, q, and r are distinct odd primes and study the torsion as well the rank of Em(Q). More specifically, we proved that if m ≢ 0 (mod 3), m ≡ 2 (mod 2k) where k ≥ 5, and none of the prime numbers p, q and r divide m, then the torsion subgroup of Em(Q) is trivial and a lower bound of the Q rank of this family of elliptic curves is 2.

In this article, we show that the quartic Diophantine equations x4 ± pqy4 = ±z2 and x4 ± pqy4 = ±iz2 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).