On Families of Elliptic Curves Ep,q:y2=x3−pqx That Intersect the Same Line La,b:y=abx of Rational Slope

We investigate a special family of elliptic curves, namely Ep,q:y2=x3−pqx, where p<q are odd primes. We study sufficient conditions for p and q so that the corresponding elliptic curve has non-trivial rational points. The number of sufficient conditions reduces to six. These six sufficient conditions relate to Polignac’s conjecture, to the prime gap problem, the twin prime conjecture, and to results from Green and Sawhney and Friedlander and Iwaniec. Additionally, we analyze the structures of the sufficient conditions for p and q by their graphical visualizations of the six sufficient conditions for p,q≤6997. The graphical structures for the six sufficient conditions exhibit arc structures, quasi-linear arc segments, tile structures, and sparsely populated structures.