For the integer D = pq of the product of two distinct odd primes, we construct an elliptic curve E2rD : y 2 = x3 − 2rDx over Q, where r is a parameter dependent on the classes of p and q modulo 8, and show, under Birch and Swinnerton-Dyer conjecture, that the elliptic curve has rank one and vp(x([k]Q)) 6= vq(x([k]Q)) for odd k and a generator Q of the free part of E2rD(Q). Thus we can get p or ...

The equation y2 = x3 +k, for k ∈ Z, is called Mordell’s equation on account of Mordell’s long interest in it throughout his life. A natural number-theoretic task is the description of all rational and integral solutions to such an equation, either qualitatively (decide if there are finitely or infinitely many solutions in Z or Q) or quantitatively (list or otherwise conveniently describe all su...

Astronomers and biologists have had telescopes and microscopes respectively to aid in their research. With the advent of the computer, mathematicians acquired a powerful tool, using which they could generate data, make conjectures and try turning them into theorems — this was the dawn of the golden age of experimental mathematics. My research sits at the crossroads of number theory, algorithms ...

let $e$ be an elliptic curve over $bbb{q}$ with the given weierstrass equation $ y^2=x^3+ax+b$. if $d$ is a squarefree integer, then let $e^{(d)}$ denote the $d$-quadratic twist of $e$ that is given by $e^{(d)}: y^2=x^3+ad^2x+bd^3$. let $e^{(d)}(bbb{q})$ be the group of $bbb{q}$-rational points of $e^{(d)}$. it is conjectured by j. silverman that there are infinitely many primes $p$ for which $...

In this paper, we give a new and direct proof for the recently proved conjecture raised in Soltani and Roozegar (2012). The conjecture can be proved in a few lines via the integral representation of the Gauss-hypergeometric function unlike the long proof in Roozegar and Soltani (2013).

We study p-indivisibility of the central values L(1, Ed) of quadratic twists Ed of a semi-stable elliptic curve E of conductorN. A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants d splits naturally into several families FS, indexed by subsets S of the primes dividing N. Let δS = gcdd∈FS L(1, Ed) alg, where L(1, Ed) alg denotes the algeb...

The abelian group E(Q) of rational points on E is finitely generated [Mor22]. Hence E(Q) ' Z ⊕ T for some nonnegative integer r (the rank) and some finite abelian group T (the torsion subgroup). The torsion subgroup is well understood, thanks to B. Mazur [Maz77], but the rank remains a mystery. Already in 1901, H. Poincaré [Poi01, p. 173] asked what is the range of possibilities for the minimum...

  In order to find an alternative source for the highly unsaturated fatty acids oil with lower cost, a marine trash fish Odonus niger has been analysed and characterised. The body weight and the corresponding liver weight of the fish were studied and the results showed that for one gram increase in the size of the animal, the liver weight can increase to 0.04 g. The oil yielding capacity of the...

Well-known analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we discuss traffic of this sort, in both directions, in the theory of elliptic curves. In the first part of the paper, we consider various works on Heegner points and Gross–Zagier formulas in the function field context; these works lead to a ...