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On families of elliptic curves $E_{p,q}:y^2=x^3-pqx$ that intersect the same line $L_{a,b}:y=\frac{a}{b}x$ of rational slope

Eldar Sultanow, Anja Jeschke, Amir Darwish Tfiha, Madjid Tehrani, William J Buchanan

TL;DR

The paper investigates when curves E_{p,q}: y^2 = x^3 − pqx, with p<q primes, intersect a fixed rational-slope line L_{a,b}: y = (a/b)x, by reducing the problem to divisibility patterns of 4pqb^4 and isolating six viable cases from sixty total divisor-splits. It then visualizes, for p,q ≤ 6997, the resulting arithmetic structures as arc, tile, or sparse patterns, providing concrete sample curves and explaining observed shapes through number-theoretic principles (e.g., primes of the form a^2 + b^4, representations of pq, and congruence constraints). The six cases connect to broader topics such as Friedlander–Iwaniec primes, sums of squares and fourth powers, and prime-gap heuristics, framing a visually-guided approach to identifying primes that yield rational points on E_{p,q}. The work highlights both promising avenues for locating positive-rank instances and the limitations imposed by deep conjectures in number theory, suggesting future exploration of fixed lines, fixed curves, and refined distributional analyses of prime pairs.

Abstract

Let $p$ and $q$ be two distinct odd primes, $p<q$ and $E_{p,q}:y^2=x^3-pqx$ be an elliptic curve. Fix a line $L_{a.b}:y=\frac{a}{b}x$ where $a\in \mathbb{Z},b\in \mathbb{N}$ and $(a,b)=1$. We study sufficient conditions that $p$ and $q$ must satisfy so that there are infinitely many elliptic curves $E_{p,q}$ that intersect $L_{a,b}$.

On families of elliptic curves $E_{p,q}:y^2=x^3-pqx$ that intersect the same line $L_{a,b}:y=\frac{a}{b}x$ of rational slope

TL;DR

The paper investigates when curves E_{p,q}: y^2 = x^3 − pqx, with p<q primes, intersect a fixed rational-slope line L_{a,b}: y = (a/b)x, by reducing the problem to divisibility patterns of 4pqb^4 and isolating six viable cases from sixty total divisor-splits. It then visualizes, for p,q ≤ 6997, the resulting arithmetic structures as arc, tile, or sparse patterns, providing concrete sample curves and explaining observed shapes through number-theoretic principles (e.g., primes of the form a^2 + b^4, representations of pq, and congruence constraints). The six cases connect to broader topics such as Friedlander–Iwaniec primes, sums of squares and fourth powers, and prime-gap heuristics, framing a visually-guided approach to identifying primes that yield rational points on E_{p,q}. The work highlights both promising avenues for locating positive-rank instances and the limitations imposed by deep conjectures in number theory, suggesting future exploration of fixed lines, fixed curves, and refined distributional analyses of prime pairs.

Abstract

Let and be two distinct odd primes, and be an elliptic curve. Fix a line where and . We study sufficient conditions that and must satisfy so that there are infinitely many elliptic curves that intersect .
Paper Structure (20 sections, 8 equations, 9 figures, 2 tables)

This paper contains 20 sections, 8 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Candidate conditions
  3. Six sufficient conditions
  4. Case 17
  5. Case 26
  6. Case 32
  7. Case 40
  8. Case 47
  9. Case 56
  10. Graphical structures
  11. Case 17: Arc structure and quasi-linear arc segment
  12. Case 26: Tile structure
  13. Case 32: Sparsely populated structure
  14. Case 40: Arc structure
  15. Case 47: Quasi-linear arc segment
  16. ...and 5 more sections

Figures (9)

  • Figure 1: Sample elliptic curves for case 40 (left) and case 56 (right) that are shown in Table \ref{['tab:six_conditions']}. The two rational points are marked in orange and blue, respectively, for both cases.
  • Figure 2: Case 17 (Sections \ref{['sec:tab1_case17']} and \ref{['sec:structure_17']}) up to $p,q \le 6997$ showing arc structures. The plot shows odd primes p as a function of odd primes q.
  • Figure 3: Case 26 (Sections \ref{['sec:tab1_case26']} and \ref{['sec:structure_26']}) up to $p,q \le 6997$ showing tile structures. The plot shows odd primes p as a function of odd primes q.
  • Figure 4: Case 32 (Sections \ref{['sec:tab1_case32']} and \ref{['sec:structure_32']}) up to $p,q \le 6997$ showing sparsely populated solution area. The plot shows odd primes p as a function of odd primes q.
  • Figure 5: Case 40 (Sections \ref{['sec:tab1_case40']} and \ref{['sec:structure_40']}) up to $p,q \le 6997$ showing arc structures. The plot shows odd primes p as a function of odd primes q.
  • ...and 4 more figures