On Families of Elliptic Curves ${\mathit{E}}_{\mathit{p}\mathit{,}\mathit{q}}\mathbf{:}{\mathit{y}}^{\mathit{2}}={\mathit{x}}^{\mathit{3}}-\mathit{p}\mathit{q}\mathit{x}$ That Intersect the Same Line ${\mathit{L}}_{\mathit{a}\mathbf{,}\mathit{b}}\mathbf{:}\mathit{y}=\genfrac{}{}{.05ex}{}{\mathit{a}}{\mathit{b}}\mathit{x}$ of Rational Slope

Abstract

We investigate a special family of elliptic curves, namely $E_{p,q}:y^2=x^3-pqx$, 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 $p,q\le 6997$. The graphical structures for the six sufficient conditions exhibit arc structures, quasi-linear arc segments, tile structures, and sparsely populated structures.

1. Introduction

The question of whether an elliptic curve has rational points or not has been occupying mathematicians for a while. The study [1] proves that there is an upper bound of the rank of this family of elliptic curves. In addition, they prove that there are sufficient conditions on p and q for the elliptic curves to have rank two, three, and four, and that there exist infinitely many such primes. Article [2] shows that the additional restriction $p\equiv q\equiv 3$ mod 16 leads to the rank of the Mordell–Weil group $E_{pq}\left(\mathbb{Q}\right)$ being less than or equal to 1, for example.

Rather than fixing a single line $L_{a,b}$, we ask whether there exists a rational-slope line intersecting $E_{p,q}$ in a non-trivial rational point. The formulation in terms of $L_{a,b}$ serves as a constructive tool to detect such points.

1.1. Rank and Torsion (Reminder)

For an elliptic curve $E/\mathbb{Q}$, the Mordell–Weil group $E\left(\mathbb{Q}\right)$ is finitely generated and decomposes as $E\left(\mathbb{Q}\right)\cong E{\left(\mathbb{Q}\right)}_{\mathrm{tor}}\oplus {\mathbb{Z}}^r$. Here, $E{\left(\mathbb{Q}\right)}_{\mathrm{tor}}$ is the torsion subgroup (points of finite order), and r is the (Mordell–Weil) rank, measuring the number of independent rational points of infinite order. A non-trivial rational point means a point $P\in E\left(\mathbb{Q}\right)$ with $P\ne \mathcal{O}$ and $y\left(P\right)\ne 0$; hence, $P\notin E{\left(\mathbb{Q}\right)}_{\mathrm{tor}}$.

1.2. Contribution and Scope

This paper contributes a data-driven reduction and visualization pipeline for the family $E_{p,q}:y^2=x^3-pqx$ with $p<q$ odd primes. We reduce 60 algebraic factor-splitting candidates to six solvable sufficient conditions for the existence of non-trivial rational points. We then map the solution sets for $p,q\le 6997$ into prime-index coordinates and classify the observed patterns (arcs, quasi-linear bands, tiles, sparsity). Links to Polignac-type statements and prime gap heuristics serve as context only; the graphical structures are empirical observations in the investigated range, while cited theorems (e.g., Friedlander–Iwaniec) remain fully proved results.

Usual directions of studies take care about the rank of elliptic curves. Our goal is to find structures of the conditions on the odd primes $p,q$ by visualization. We concentrate on sufficient conditions on the odd primes $p,q$, such that the corresponding elliptic curve has non-trivial rational points.

2. Candidate Conditions

Throughout the paper, the parameters $(a,b)$ are allowed to vary. Our goal is to determine conditions on $(p,q)$ under which at least one such line $L_{a,b}$ yields a rational intersection point.

Let $L_{a,b}:y={\displaystyle \frac{a}{b}}x$ be a linear function of rational slope where $a\in \mathbb{Z},b\in \mathbb{N}$, and $(a,b)=1$. Let $E_{p,q}:y^2=x^3-pqx$ be an elliptic curve where $p<q$ are fixed odd primes. We aim to find sufficient conditions for p and q so that there are non-trivial rational points on the elliptic curve $E_{p,q}$. Inserting $L_{a,b}$ into $E_{p,q}$ leads to

$$x^3-{\left({\displaystyle \frac{a}{b}}\right)}^2{x}^2-pqx=0.$$

This cubic function has two non-trivial solutions. They are given by

$$x_{1,2}={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{a}{b}}\right)}^2\pm {\displaystyle \frac{1}{2}}\sqrt{{\left({\displaystyle \frac{a}{b}}\right)}^4+4pq}.$$

In order for $x_{1,2}$ to be rational, we need $a^4+4pq{b}^4$ to be a square:

$$a^4+4pq{b}^4=c^2$$

for some $c\in \mathbb{N}$. This equation can be factored as $4pq{b}^4=c^2-a^4=(c-a^2)(c+a^2)$. We want to find all possibilities of how to separate the factors of $4pq{b}^4$ into the two factors $c-a^2$ and $c+a^2$.

Counting the number of cases boils down to computing the number of divisors of $4pq{b}^4$. For $\tau \left(n\right)$, the divisor function with $n=p_1^{e_1}\cdots p_k^{e_k}$ and $p_k$ prime, we have the identity $\tau \left(n\right)=(e_1+1)\cdots (e_k+1)$; see [3] (p. 34). If b is prime, the number of cases would be

$$\tau \left(2^{2}pq{b}^4\right)=(2+1)(1+1)(1+1)(4+1)=60.$$

If b is not prime, the number of cases would be greater. We do not assume b to be prime, but we will keep b as a separate factor to keep the number of cases to consider at 60.

Table 1. Conditions on p and q (case 1–30) derived from 60 possible cases to split the term $4pq{b}^4$ into two factors $(c+a^2)$ and $(c-a^2)$.

Case	$\mathit{c}-{\mathit{a}}^2$	$\mathit{c}+{\mathit{a}}^2$	Condition
1	$4pq{b}^4$	1	$pq={\displaystyle \frac{1-2a^2}{4b^4}}$
2	$2pq{b}^4$	2	$pq={\displaystyle \frac{1-a^2}{b^4}}$
3	$pq{b}^4$	4	$pq={\displaystyle \frac{4-2a^2}{b^4}}$
4	$b^4$	$4pq$	$pq={\displaystyle \frac{2a^2+b^4}{4}}$
5	$b^3$	$4pqb$	$pq={\displaystyle \frac{2a^2+b^3}{4b}}$
6	$b^2$	$4pq{b}^2$	$pq={\displaystyle \frac{2a^2+b^2}{4b^2}}$
7	b	$4pq{b}^3$	$pq={\displaystyle \frac{2a^2+b}{4b^3}}$
8	$2pb$	$2q{b}^3$	$p=q{b}^2-{\displaystyle \frac{a^2}{b}}$
9	$4q{b}^4$	p	$p=2a^2+4q{b}^4$
10	$2q{b}^4$	$2p$	$p=a^2+q{b}^4$
11	$q{b}^4$	$4p$	$p={\displaystyle \frac{2a^2+q{b}^4}{4}}$
12	$q{b}^3$	$4pb$	$p={\displaystyle \frac{2a^2+q{b}^3}{4b}}$
13	$q{b}^2$	$4p{b}^2$	$p={\displaystyle \frac{2a^2+q{b}^2}{4b^2}}$
14	$qb$	$4p{b}^3$	$p={\displaystyle \frac{2a^2+qb}{4b^3}}$
15	$2qb$	$2p{b}^3$	$q=p{b}^2-{\displaystyle \frac{a^2}{b}}$
16	$4p{b}^4$	q	$q=2a^2+4p{b}^4$
17	$2p{b}^4$	$2q$	$q=a^2+p{b}^4$
18	$p{b}^4$	$4q$	$q={\displaystyle \frac{2a^2+p{b}^4}{4}}$
19	$p{b}^3$	$4qb$	$q={\displaystyle \frac{2a^2+p{b}^3}{4b}}$
20	$p{b}^2$	$4q{b}^2$	$q={\displaystyle \frac{2a^2+p{b}^2}{4b^2}}$
21	$pb$	$4q{b}^3$	$q={\displaystyle \frac{2a^2+pb}{4b^3}}$
22	$2q{b}^2$	$2p{b}^2$	$p=q+{\displaystyle \frac{a^2}{b^2}}$
23	$2b$	$2pq{b}^3$	$pq={\displaystyle \frac{a^2+b}{b^3}}$
24	$2b^2$	$2pq{b}^2$	$pq={\displaystyle \frac{a^2+b^2}{b^2}}$
25	$2b^3$	$2pqb$	$pq={\displaystyle \frac{a^2+b^3}{b}}$
26	$2b^4$	$2pq$	$pq=a^2+b^4$
27	$4b$	$pq{b}^3$	$pq={\displaystyle \frac{2a^2+4b}{b^3}}$
28	$4b^2$	$pq{b}^2$	$pq={\displaystyle \frac{2a^2+4b^2}{b^2}}$
29	$4b^3$	$pqb$	$pq={\displaystyle \frac{2a^2+4b^3}{b}}$
30	$4b^4$	$pq$	$pq=2a^2+4b^4$

presents all 60 cases and their resulting conditions for p and q.

3. Six Sufficient Conditions

The 60 candidate cases in Table 1 and

Table 2. Conditions on p and q (case 31–60) derived from 60 possible cases to split the term $4pq{b}^4$ into two factors, $(c+a^2)$ and $(c-a^2)$.

Case	$\mathit{c}-{\mathit{a}}^2$	$\mathit{c}+{\mathit{a}}^2$	Condition
31	1	$4pq{b}^4$	$pq={\displaystyle \frac{2a^2+1}{4b^4}}$
32	2	$2pq{b}^4$	$pq={\displaystyle \frac{a^2+1}{b^4}}$
33	4	$pq{b}^4$	$pq={\displaystyle \frac{2a^2+4}{b^4}}$
34	$4pq$	$b^4$	$pq={\displaystyle \frac{b^4-2a^2}{4}}$
35	$4pqb$	$b^3$	$pq={\displaystyle \frac{b^3-2a^2}{4b}}$
36	$4pq{b}^2$	$b^2$	$pq={\displaystyle \frac{b^2-2a^2}{4b^2}}$
37	$4pq{b}^3$	b	$pq={\displaystyle \frac{b-2a^2}{4b^3}}$
38	$2q{b}^3$	$2pb$	$p=q{b}^2+{\displaystyle \frac{a^2}{b}}$
39	p	$4q{b}^4$	$p=4q{b}^4-2a^2$
40	$2p$	$2q{b}^4$	$p=q{b}^4-a^2$
41	$4p$	$q{b}^4$	$p={\displaystyle \frac{q{b}^4-2a^2}{4}}$
42	$4pb$	$q{b}^3$	$p={\displaystyle \frac{q{b}^3-2a^2}{4b}}$
43	$4p{b}^2$	$q{b}^2$	$p={\displaystyle \frac{q{b}^2-2a^2}{4b^2}}$
44	$4p{b}^3$	$qb$	$p={\displaystyle \frac{qb-2a^2}{4b^3}}$
45	$2p{b}^3$	$2qb$	$q=p{b}^2+{\displaystyle \frac{a^2}{b}}$
46	q	$4p{b}^4$	$q=4p{b}^4-2a^2$
47	$2q$	$2p{b}^4$	$q=p{b}^4-a^2$
48	$4q$	$p{b}^4$	$q={\displaystyle \frac{p{b}^4-2a^2}{4}}$
49	$4qb$	$p{b}^3$	$q={\displaystyle \frac{p{b}^3-2a^2}{4b}}$
50	$4q{b}^2$	$p{b}^2$	$q={\displaystyle \frac{p{b}^2-2a^2}{4b^2}}$
51	$4q{b}^3$	$pb$	$q={\displaystyle \frac{pb-2a^2}{4b^3}}$
52	$2q{b}^2$	$2p{b}^2$	$p=q-{\displaystyle \frac{a^2}{b^2}}$
53	$2pq{b}^3$	$2b$	$pq={\displaystyle \frac{b-a^2}{b^3}}$
54	$2pq{b}^2$	$2b^2$	$pq={\displaystyle \frac{b^2-a^2}{b^2}}$
55	$2pqb$	$2b^3$	$pq={\displaystyle \frac{b^3-a^2}{b}}$
56	$2pq$	$2b^4$	$pq=b^4-a^2$
57	$pq{b}^3$	$4b$	$pq={\displaystyle \frac{4b-2a^2}{b^3}}$
58	$pq{b}^2$	$4b^2$	$pq={\displaystyle \frac{4b^2-2a^2}{b^2}}$
59	$pqb$	$4b^3$	$pq={\displaystyle \frac{4b^3-2a^2}{b}}$
60	$pq$	$4b^4$	$pq=4b^4-2a^2$

reduce to only six different cases that have solutions. The other 54 cases either have no solution (see Appendix A) or are redundant to one of the six cases (see Appendix B). These six cases and their resulting corresponding conditions for p and q are listed in

Table 3. Sample elliptic curves including their LMFDB reference (if existing) and their rational points for all six cases that have solutions.

Case	Condition	Samples a, b, p, q	Resulting Sample Elliptic Curve	Resulting Rational Points on Sample Elliptic Curve
17	$q=a^2+p{b}^4$	$8,3,3,307$	$y^2=x^3-921x$	$\left({\displaystyle \frac{307}{9}},{\displaystyle \frac{2456}{27}}\right),(-27,-72)$
26	$pq=a^2+b^4$	$7,2,p,{\displaystyle \frac{65}{p}}$	$y^2=x^3-65x$ [4]	$\left({\displaystyle \frac{65}{4}},{\displaystyle \frac{455}{8}}\right),(-4,-14)$
32	$pq={\displaystyle \frac{a^2+1}{b^4}}$	$1432,5,p,{\displaystyle \frac{3281}{p}}$	$y^2=x^3-3281x$	$\left(-{\displaystyle \frac{1}{25}},-{\displaystyle \frac{1432}{125}}\right)$, (82025, 23491960)
40	$p=q{b}^4-a^2$	$2,1,3,7$	$y^2=x^3-21x$ [5]	$(7,14),(-3,-6)$
47	$q=p{b}^4-a^2$	$5,2,3,23$	$y^2=x^3-69x$ [6]	$(12,30),\left(-{\displaystyle \frac{23}{4}},-{\displaystyle \frac{115}{8}}\right)$
56	$pq=b^4-a^2$	$1,2,p,{\displaystyle \frac{15}{p}}$	$y^2=x^3-15x$ [7]	$(4,2),\left(-{\displaystyle \frac{15}{4}},-{\displaystyle \frac{15}{8}}\right)$

. Table 3 also displays a sample combination of $a,b,p,q$, the resulting example elliptic curve, and their rational points. As an illustration, Figure 1 shows the sample elliptic curves of cases 40 and 56 together with their rational points.

In the following, we give some additional details about the conditions of the six cases. We do not elaborate extensively because this is not the main focus of the paper. We give some links to known results in the literature, e.g., Polignac’s conjecture [8], the prime gap problem [9], the twin prime conjecture [10,11], and results from Green and Sawhney [12] and Friedlander and Iwaniec [13].

To provide a compact overview and to distinguish between proven results, conjectural links, and empirical observations,

Table 4. Summary of the six solvable cases, their number-theoretic context, and observed visual patterns.

Case	Condition	Number-Theoretic Link	Status	Visual Pattern
17	$q=a^2+p{b}^4$	prime gaps	conjectural	arcs + quasi-linear
26	$pq=a^2+b^4$	$a^2+b^4$ primes	proved	tile-like gaps
32	$pq=(a^2+1)/b^4$	$a^2+1$ factorization	empirical	sparse
40	$p=q{b}^4-a^2$	as Case 17 (swapped)	conjectural	arcs
47	$q=p{b}^4-a^2$	$b=2$: $q\approx 16p$	empirical	quasi-linear arc
56	$pq=b^4-a^2$	prime gaps ($p+q$)	conjectural	arcs

summarizes how each of the six solvable cases relates to known number-theoretic results and how it manifests in the visualizations.

3.1. Details of Case 17: $q=a^2+p{b}^4$

If b is odd, the square $a^2$ is even, because p and q are odd. Hence, a must be even. It is $(q-p{b}^4)$ mod 4 = 0. The special case $b=1$ leads to $(q-p)$ mod 4 = 0. According to Polignac’s conjecture [8] (which is not proved), there are infinitely many cases of two consecutive prime numbers with difference n if n is even [14] (p. 295).

3.2. Details of Case 26: $pq=a^2+b^4$

There are infinitely many primes of the form $p=a^2+b^4$, according to the theorem of Friedlander and Iwaniec [13]. This result provides important context, but it does not imply that a generic semiprime $pq$ admits a representation of the form $pq=a^2+b^4$. Friedlander and Iwaniec conjectured that the sum $a^2+4b^2$ also produces an infinite number of primes if a and b are prime numbers. Proving this is exactly what Green and Sawhney have achieved [12]. Up to a big integer X, there are Y primes such that the product of two of them yields a semiprime of the form $a^2+b^4$. This observation aligns with Heath–Brown’s general result on binary forms $G(x,y)$ of degree d, showing that the number of positive integers $n<X$ representable by such forms is of exact order $X^{2/d}$, provided $G(1,0)>0$ [15] (Theorem 8). For the specific case $G(x,y)=x^2+y^4$ (i.e., $d=4$), this implies a growth rate proportional to $X^{1/2}$, consistent with the density behavior established by Friedlander and Iwaniec [13].

If $p\equiv 3$ (mod 4) or $q\equiv 3$ (mod 4), there is no solution [16] (p. 21). Set $p=r^2+s^2$ and $q=u^2+v^2$. If $p\equiv 1$ (mod 4) and $q\equiv 1$ (mod 4), we obtain exactly one solution $r>s>0$ for p and one solution $u>v>0$ for q [16] (p. 21). Hence, the product of both primes is $pq=(r^2+s^2)(u^2+v^2)={(ru+sv)}^2+{(rv-su)}^2={(ru-sv)}^2+{(rv+su)}^2$. Consider $b^2=c$, other integer solutions for $pq=a^2+c^2=a^2+b^4$ do not exist, unless one of the four integers $ru+sv$, $|rv-su|$, $|ru-sv|$, and $rv+su$ is a perfect square.

3.3. Details of Case 32: $pq={\displaystyle \frac{a^2+1}{b^4}}$

Every odd prime is of the form $4n+1$ or $4n+3$, i.e., every odd prime is congruent to 1 or to $-1$ (mod 4); see [17] (p. 72), [18] (p. 98). The product of two odd prime numbers satisfies the following modular property:

• If both primes satisfy $p\equiv 1$ (mod 4) and $q\equiv 1$ (mod 4), then their product is congruent with $p·q\equiv 1$ (mod 4).
• If one prime satisfies $p\equiv 1$ (mod 4) and the other $q\equiv 3$ (mod 4), then $p·q\equiv 3$ (mod 4).
• If both primes satisfy $p\equiv 3$ (mod 4) and $q\equiv 3$ (mod 4), then $p·q\equiv 1$ (mod 4).

Thus, the product of two odd prime numbers is 1 (mod 4) or 3 (mod 4), depending on the residue classes of both prime numbers. The square of any even number is congruent to 0 (mod 4), while the squares of any odd number are congruent to 1 (mod 4); see [16] (p. 21). Thus, quadratic residue modulo 4 can only be 0 or 1. Considering the fourth power of b, we observe the following:

$$b^4\equiv \left\{\begin{array}{cc}0\left(\mathrm{mod}4\right),\hfill & \mathrm{if}b\equiv 0,2\left(\mathrm{mod}4\right),\hfill \\ 1\left(\mathrm{mod}4\right),\hfill & \mathrm{if}b\equiv 1,3\left(\mathrm{mod}4\right).\hfill \end{array}\right.$$

Substituting this into the condition of case 32, we conclude: If $b^4\equiv 0\left(\mathrm{mod}4\right)$, then $a^2\equiv 3\left(\mathrm{mod}4\right)$, which is not possible. If $b^4\equiv 1\left(\mathrm{mod}4\right)$, then $a^2\equiv 0\left(\mathrm{mod}4\right)$, which means $a\equiv 0,2\left(\mathrm{mod}4\right)$. This ensures that a and b are systematically constrained based on their modular properties.

3.4. Details of Case 40: $p=q{b}^4-a^2$

If b is odd, the square $a^2$ is even, because p and q are odd. Hence, a must be even. It is $(q{b}^4-p)\mathrm{mod}4=0$. The special case $b=1$ results in $(q-p)\mathrm{mod}4=0$, which is the same as case 17 with $b=1$. Hence, according to Polignac’s conjecture [8] (which is not proved), there are infinitely many cases of two consecutive prime numbers with the difference n if n is even [14] (p. 295). If b is even, it is $2^4|(p+a^2)$, which can be transformed to the problem of describing primes of the form $x^2+n{y}^2$. This was extensively elaborated on by David A. Cox [19].

3.5. Details of Case 47: $q=p{b}^4-a^2$

If b is odd, the square $a^2$ is even, because p and q are odd. Hence, a must be even. It is $(p{b}^4-q)\mathrm{mod}4=0$. The special case $b=1$ has no solution because it contradicts $q>p$.

3.6. Details of Case 56: $pq=b^4-a^2$

We can write $pq=(b^2-a)(b^2+a)$, which leads to $p=b^2-a$ and $q=b^2+a$ because $p<q$, assuming $a\in \mathbb{N}$ without loss of generality. Hence, $q-p=2a$. This is a special case of the prime gap problem. According to Polignac’s conjecture [8] (which is not proved), there are infinitely many cases of two consecutive prime numbers with the difference n if n is even [14] (p. 295). If $a=1$, this is well known as the twin prime conjecture [10,11].

3.7. Remark (Sufficiency of the Six Cases)

The six cases listed in Table 3 provide sufficient conditions for the existence of non-trivial rational points on the elliptic curve $E_{p,q}:y^2=x^3-pqx$. More precisely, if odd primes $p<q$ satisfy one of these conditions, then coprime integers $a\in \mathbb{Z}$ and $b\in \mathbb{N}$ can be constructed such that the line $L_{a,b}:y={\displaystyle \frac{a}{b}}x$ intersects $E_{p,q}$ at a rational point with $y\ne 0$.

We do not claim that these conditions are necessary, nor that they characterize all elliptic curves $E_{p,q}$ admitting non-trivial rational points.

4. Graphical Structures

Using the six cases presented in Section 3, we cover curves $E_{p,q}:y^2=x^3-pqx$, $p<q$ odd primes, by searching for tuples $(a,b)$, where $a\in \mathbb{Z},b\in \mathbb{N}$, and $(a,b)=1$. In this search, we consider the finite range $p,q\le 6997$, and we let b range from 1 to 200. For each prime pair $(p,q)$ and each case, we return the first hit found for a that meets the condition. Accordingly, the visual patterns reported below are observations within this finite range and this search strategy. The detailed code to visualize the structures is on GitHub [20].

The pictures in this section should be read as follows: The quadratic structure is stored in matrix format. The entry in row n and column m is marked in yellow if there exists an $a\in \mathbb{Z},b\in \mathbb{N}$, $(a,b)=1$ satisfying the condition of the case considered, e.g., $q=a^2+p{b}^4$, where p is the $(n+1)$-th prime number and q is the $(m+1)$-th prime number. The shift in the index occurs because we only consider odd primes for plotting. This means $n+1=\pi \left(p\right)$, $m+1=\pi \left(q\right)$. To improve readability, we have labeled the matrix axes with the actual prime numbers instead of their indices. Hence, the axes in all figures contain only odd primes. The plots show all odd primes p as a function of odd primes q, considering primes $p,q<6997$.

The conditions of the cases are stated in the natural coordinates $(p,q)\in {\mathbb{N}}^2$. For plotting, we use prime-index coordinates $\left(\pi \right(p),\pi (q\left)\right)$, where $\pi (·)$ denotes the prime counting function restricted to odd primes. This change of coordinates distorts straight lines into gently curved arcs in figures of cases 17, 40, 47, and 56. By the prime number theorem [21,22], the i-th prime satisfies $p_i\sim ilogi$, hence, $p\sim \pi \left(p\right)log\pi \left(p\right)$ and similarly for q. Because $xlogx$ is locally curved, affine relations in $(p,q)$ appear as arcs in $\left(\pi \right(p),\pi (q\left)\right)$. In the finite plotting window $p,q\le 6997$, the relative scaling is well approximated by $log\left(p_{max}\right)/log\left(q_{max}\right)$, where $p_{max},q_{max}$ are the maximal plotted values $p,q$ in each figure.

The six cases reveal different structures. These structures are plotted and analyzed in more detail in the following subsections.

4.1. Case 17: Arc Structure and Quasi-Linear Arc Segment

Figure 2 displays the graphical structure of case 17 with its condition $q=a^2+p{b}^4$. For fixed b and moderately small $\left|a\right|$, the solutions lie close to the affine lines $q\approx b^{4}p\left(\mathrm{up}\mathrm{to}\mathrm{the}\mathrm{shift}{a}^2\right)$ in the $(p,q)$-plane. These affine lines appear as arcs in Figure 2 due to the prime number coordinates used for plotting: Using the prime number theorem, the line $q\approx b^{4}p$ transforms approximately into $\pi \left(q\right)log\pi \left(q\right)\approx b^4\pi \left(p\right)log\pi \left(p\right)$, which is slightly curved in prime number coordinates.

The parameter b controls the “slope”: $b=1$ yields arcs hugging the diagonal $q=p$, while larger b push the arcs above that diagonal. The finite cut-off $p,q\le 6997$ truncates these arcs to visible segments. Modular constraints described in Section 3.1 (e.g., $q-p{b}^4\equiv a^2\left(\mathrm{mod}4\right)$ with a even when b is odd) thin out these bands and create small lacunae along the arcs, exactly as seen in Figure 2.

In addition to these arc structures, a quasi-linear arc segment $q\approx 16p$ is visible for lower values of p. It is explained by setting $b=2$ in the condition for moderately small $\left|a\right|$. A similar quasi-linear arc segment is visible in case 47, too. We refer to Section 4.5, dealing with case 47 for a more detailed discussion.

4.2. Case 26: Tile Structure

Figure 3 displays the graphical structure of case 26 with its condition $pq=a^2+b^4$. The tile structure shines through very slightly in Figure 3. Such tiles may be due to the fact that for a certain prime number p or q, the product $pq$ cannot be represented as a sum of $a^2+b^4$, or only with great difficulty. In detail, this phenomenon can be linked to deeper number-theoretic conjectures about representations of integers as sums of powers. In particular, the theorem from Friedlander and Iwaniec [13] shows that there are infinitely many primes of the form $p=a^2+b^4$, but it does not guarantee that every product $pq$ of two primes can be expressed as $a^2+b^4$ for integers a and b. This structural limitation hints at the sparsity of solutions in the matrix, resulting in an almost checkerboard-like appearance. Furthermore, such expressions can also be viewed through the lens of the Hardy–Littlewood circle method [23], which predicts that representations of numbers as sums of squares and higher powers are often rare due to cancellation phenomena in exponential sums. Moreover, the Bunyakovsky conjecture [24] states that certain irreducible polynomials, such as $a^2+b^4$, should produce primes infinitely often under certain conditions, but it does not cover composite products $pq$. Therefore, for many prime pairs $(p,q)$, there may simply be no integers a and b satisfying $pq=a^2+b^4$, which explains the observed gaps and tile structures in the visualization. This interpretation is heuristic and based on the plotted data in the finite range $p,q\le 6997$. We do not claim a proven characterization of when a semiprime $pq$ is representable as $a^2+b^4$.

4.3. Case 32: Sparsely Populated Structure

Figure 4 displays the graphical structure of case 32. Solutions to the condition of case 32, namely $pq={\displaystyle \frac{a^2+1}{b^4}}$, are rare: No solution has been discovered with all p, q, a, b being small. However, some small values of p and q appear in the graphical representation when at least one of the values of a and b is large. For example, a yellow dot in the image that is close to $(p,q)=(3,3)$ corresponds to $(p,q)=(5,337)$ with $(a,b)=$ (34522, 29). The smallest example for $b>1$ is $y^2=x^3-3281x$, which is the sample elliptic curve for case 32 shown in Table 3. This rarity can be explained by the factorization properties of expressions of the form $x^2+1$. A well-known related case is the factorization of Fermat numbers, given by $F_n=2^{2^n}+1$. It is well known that every Fermat number $F_n$ is prime or has at least one very large prime factor [25,26]. This illustrates a general principle: Expressions of the form $x^2+1$ tend to have a very restricted set of prime factors or only a few, making it difficult for them to satisfy a product condition involving two small primes. Furthermore, the Bunyakovsky conjecture [24] on prime values of polynomials suggests that certain irreducible polynomials, such as $x^2+1$, take prime values only for a limited set of integers. Although there is no general proof of the conjecture, its implications suggest that for most values of a, the expression $a^2+1$ is unlikely to split into exactly two small prime factors, further restricting possible solutions to the given equation. These constraints, combined with the modular properties described in Section 3.3, explain why valid pairs $(p,q)$ are extremely rare. The modular constraints above rule out broad classes of parameters, but they do not by themselves predict how often $a^2+1$ factors as $pq{b}^4$ with primes $p<q$. In our finite range $p,q\le 6997$, we observe that valid pairs $(p,q)$ are extremely rare and typically require large $\left|a\right|$ and/or b. This observation is empirical and its explanation is necessarily heuristic and based on the plotted data.

4.4. Case 40: Arc Structure

Figure 5 displays the graphical structure of case 40 with its condition $p=q{b}^4-a^2$. The structure of cases 17 and 40 are similar, as the analysis in Section 3.1 and Section 3.4 shows. For fixed b and small $\left|a\right|$, valid pairs concentrate near the straight lines $p\approx b^{4}q(\mathrm{again}\mathrm{up}\mathrm{to}\mathrm{the}\mathrm{shift}-a^2)$ in the $(p,q)$-plane. When plotting in prime number coordinates, these lines bend into arcs for the same reason as in case 17: Using the prime number theorem, the line $p\approx b^{4}q$ becomes $\pi \left(p\right)log\pi \left(p\right)\approx b^4\pi \left(q\right)log\pi \left(q\right)$, producing gently curved arc structures in Figure 5.

The parity conditions summarized in Section 3.4 ($q{b}^4-p\equiv 0\left(\mathrm{mod}4\right)$ and, for even b, the reduction to primes represented by $x^2+n{y}^2$) further sparsifies these arcs, explaining the banded but porous appearance of the solution set. As b increases, the main arcs shift away from the diagonal and become harder to distinguish within the plotting window $p,q\le 6997$, leaving only the lower-b branches prominently visible.

4.5. Case 47: Quasi-Linear Arc Segment

Figure 6 displays the graphical structure of case 47 with its condition $q=p{b}^4-a^2$, reminiscent of a rotating pointer line on a submarine radar screen: We see a quasi-linear arc segment following the simple relation $q\approx 16p$ [27] for lower values of p. Figure 7 shows the same visualization, but the quasi-linear arc segment highlighted in red. The quasi-linear arc segment is explained by setting $b=2$ in the condition, which leads to $q=16p-a^2$. For small values of $\left|a\right|$, this simplifies to $q\approx 16p$, which means that valid prime pairs tend to accumulate on this quasi-linear arc segment or below this quasi-linear arc segment. In our finite range $p,q\le 6997$, we observe this band most clearly for $b=2$; higher-b bands are expected but are difficult to distinguish at this scale. This interpretation is heuristic and based on the plotted data.

Considering prime number coordinates, the slope of the quasi-linear arc segment in Figure 6 is approximately

$$\begin{array}{c}\hfill {\displaystyle \frac{\pi \left(p_{\max }\right)}{\pi \left(q_{\max }\right)}}\approx {\displaystyle \frac{\pi \left(\frac{1}{16}{q}_{\max }\right)}{\pi \left(q_{\max }\right)}}={\displaystyle \frac{\pi \left(437\right)}{\pi \left(6997\right)}}={\displaystyle \frac{84}{900}}\approx 0.095.\end{array}$$

Hence, the fitted red line in the visualized data range up to $p_{\max },q_{\max }<6997$ in Figure 7 is $\pi \left(p\right)\approx 10.5\pi \left(q\right)$ instead of $q\approx 16p$. The transformation factor between the two coordinate systems is

$$\begin{array}{c}\hfill {\displaystyle \frac{log\left(p_{\max }\right)}{log\left(q_{\max }\right)}}\approx {\displaystyle \frac{log\left(\frac{1}{16}{q}_{\max }\right)}{log\left(q_{\max }\right)}}={\displaystyle \frac{log\left(\frac{1}{16}\right)+log\left(q_{\max }\right)}{log\left(q_{\max }\right)}}\approx {\displaystyle \frac{-1.2+3.84}{3.84}}\approx 0.69\approx {\displaystyle \frac{10.5}{16}}.\end{array}$$

When looking at larger numbers $(p_{\max },q_{\max })$, the transformation factor ${\displaystyle \frac{log\left(p_{\max }\right)}{log\left(q_{\max }\right)}}$ will approach 1, and thus, the quasi-linear arc segment will gradually curve toward the slope ${\displaystyle \frac{1}{16}}$ [27]. Hence, the quasi-linear arc segment in Figure 6 is a small extract of arc structures, similar to the observations with regard to cases 17 and 40.

Additional quasi-linear arc segment structures should emerge for higher values of $b\in \mathbb{N}$. For $b=3$ and $b=4$, for example, we expect quasi-linear arc segments with $q\approx 81p$ and $q\approx 256p$, etc. These structures should be observable above the quasi-linear arc segment for $b=2$ by analyzing a sufficiently large dataset concerning p and q. Our choice $p,q<6997$ makes these additional quasi-linear arc segment structures hard to distinguish from each other, although Figure 6 shows a slight increase in the density of valid prime pairs in the area $p<6997/16\approx <437$. This becomes more apparent under logarithmic scaling or higher contrast, suggesting a deeper regularity in how prime numbers align under this equation and motivating further investigation into their distributional properties.

4.6. Case 56: Arc Structure

Figure 8 displays the graphical structure of case 56 with its condition $pq=b^4-a^2$, showing slightly bent arcs. To investigate further, we extracted one of these arcs and found that a simple circle equation ${(x-x_0)}^2+{(y-y_0)}^2=R^2$ with a fixed center $x_0,y_0$ and radius R captures almost all points of such an arc. For example, let us choose $x_0=-1051,y_0=-1051,R=1700$. We almost perfectly capture a significant portion of these points as highlighted in red in Figure 9. The numerical values of the fitted circle (center $(-1051,-1051)$ and radius 1700) have no intrinsic arithmetic meaning. They arise from a local geometric approximation of the curved image of the linear relation $p+q=2b^2$ under the prime-index transformation and depend on the chosen plotting range.

This visual phenomenon can be explained straightforwardly by taking a closer look [28] at condition $pq=b^4-a^2$ of case 56: This condition can be factored as $b^4-a^2=(b^2+a)(b^2-a)$, which suggests that for prime numbers p and q, it is $q=b^2+a$ and $p=b^2-a$ due to $p<q$. By rearranging, we obtain

$$a={\displaystyle \frac{q-p}{2}},b=\sqrt{{\displaystyle \frac{p+q}{2}}}.$$

Thus, solutions exist when ${\displaystyle \frac{p+q}{2}}$ is a perfect square. These solutions lie along the lines $p+q=2b^2$ in the coordinates $(p,q)\in {\mathbb{N}}^2$. Plotting in prime number coordinates distorts the lines to arcs: Considering the prime number theorem, the lines $p+q=2b^2$ transform to $\pi \left(p\right)log\pi \left(p\right)+\pi \left(q\right)log\pi \left(q\right)=2b^2$, generating slightly curved arc structures in Figure 8. In our finite range $p,q\le 6997$, we observe several such arc bands corresponding to different values of b. This interpretation is heuristic and based on the plotted data.

5. Discussion and Outlook

So far, we have inferred the conditions that two distinct odd primes $p,q$, $p<q$ must satisfy so that the elliptic curve $E_{p,q}:y^2=x^3-pqx$ can have rational points. There exist many curves that fulfill at least one of the six conditions and therefore have rational points. Similar curves of the form $E_n:y^2=x^3-n^{2}x$ with $n\ne 0$ as a rational number are well known and exhibit the following properties:

• The rational number n is a congruent number if and only if there is a point $P=(x,y)\in E_n\left(\mathbb{Q}\right)$ with $y\ne 0$; see [29] (169), [30] (144).
• The only rational points of finite order in $E_n$ are four points of order 2, namely $\mathcal{O}$ (the point at infinity), $(0,0)$, and $(\pm n,0)$; see [31] (44). In other words, $E_n{\left(\mathbb{Q}\right)}_{\mathrm{tors}}=\{\mathcal{O},(0,0),(n,0),(-n,0)\}$.

Combining these two statements leads to the conclusion that the rational number n is a congruent number if and only if the elliptic curve $E_n=y^2=x^3-n^{2}x$ has a positive rank. This is true because the existence of a point $P=(x,y)\in E_n\left(\mathbb{Q}\right)$ with $y\ne 0$ is equivalent to P is not in $E_n{\left(\mathbb{Q}\right)}_{\mathrm{tors}}$. Hence, P has infinite order, and in addition, the elliptic curve has a positive rank. Researchers have also already addressed congruent numbers with many prime factors using the curve $y^2=x^3-n^{2}x$; see for example [32]. Open questions with regard to the elliptic curve $E_{p,q}:y^2=x^3-pqx$ are, e.g., as follows:

• What must the criteria for p and q be for this family of elliptic curves to have a positive rank?
• Under which circumstances for given primes $p,q$ are one of our six conditions sure to yield an elliptic curve with a positive rank?

The criterion that $p<q$ are odd primes is not sufficient because two counterexamples are $p=5,q=7$ and $p=3,q=11$ (see

Table 5. Two rank-zero counterexamples from LMFDB used to show that $p<q$ odd primes alone do not ensure a positive rank.

Curve	Simplified Model	Rank r	$\mathit{E}{\left(\mathbb{Q}\right)}_{\mathit{tor}}$
LMFDB 78400.ga1 [33]	$y^2=x^3-35x$	0	$\mathbb{Z}/2\mathbb{Z}$
LMFDB 34848.ci1 [34]	$y^2=x^3-33x$	0	$\mathbb{Z}/2\mathbb{Z}$

). The corresponding elliptic curves have rank zero, and their torsion generators are trivial; see LMFDB [33,34]:

These questions may be answered in future investigations. Further interesting avenues for future research include the following:

• Fixing a line and exploring families of curves where $pq$ is a congruent number. Visualizing the corresponding $(p,q)$ pairs may reveal underlying structures.
• Fixing a curve and investigating families of lines that intersect it at rational points, potentially uncovering deeper arithmetic properties.
• Analyzing the visualized structures in greater detail and interpreting them to draw meaningful conclusions about the distribution of prime numbers.