Congruent Number Elliptic Curves Related to Integral Solutions of m^{2} = n^{2} + nl + n^{2}

Lorenz Halbeisen and Norbert Hungerbühler Department of Mathematics ETH Zentrum 8092 Zürich Switzerland

                 lorenz.halbeisen@math.ethz.ch
              norbert.hungerbuehler@math.ethz.ch

Abstract

We construct an infinite family of congruent number elliptic curves, each with rank at least two, which are related to integral solutions of m^{2} = n^{2} + nl + l^{2}.

1 Introduction

Elliptic curves and their geometric and algebraic structure have been a flourishing field of research in the past. They find prominent applications in cryptography and played a key role in the proof of Fermat’s Last Theorem. A salient feature of the algebraic structure of an elliptic curve is its rank. Among general elliptic curves, congruent number curves of high rank are of particular interest (see, for example, [2]). More difficult than finding an individual congruent number curve of high rank is to find infinite families of such curves. Johnstone and Spearman [7] constructed such a family with rank at least three, which is related to rational points on the biquadratic curve w^{2} = t^{4} + 14t^{2} + 4. In the present paper, we show an elementary construction for an infinite family of congruent number curves of rank at least two that are related to the quadratic diophantine equation m^{2} = n^{2} + nl + l^{2}, and which have three integral points with positive y-coordinate on a straight line. Incidentally, some members of the family exhibit surprisingly high individual rank, namely rank five (whereas the members of the family given by Johnstone and Spearman [7] all have rank three). We start by recalling some basic results on congruent numbers.

A positive integer A is called a congruent number if A is the area of a right-angled triangle with three rational sides. So A is congruent if and only if there exists a rational Pythagorean triple (a, b, c) (i.e., a, b, c ∈ ℚ, a^{2} + b^{2} = c^{2}, and ab ≠ 0) such that \frac{ab}{2} = A. The sequence of integer congruent numbers starts with

5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, …

(see, for example, the On-Line Encyclopedia of Integer Sequences [10, \underline{A003273}]). It is well-known that A is a congruent number if and only if the cubic curve

C_{A} : y^{2} = x^{3} − A^{2}x

has a rational point (x_{0}, y_{0}) with y_{0} ≠ 0. The cubic curve C_{A} is called a congruent number elliptic curve or just congruent number curve. With respect to some congruent number A, the correspondence between rational points (x, y) with y ≠ 0 on the congruent number curve C_{A} on the one hand, and rational Pythagorean triples (a, b, c) with ab = 2A on the other hand, is given by

(a, b, c) ↦ (\frac{b(b + c)}{2} , \frac{b^{2}(b + c)}{2}),  (1)

and

(x, y) ↦ (\frac{2xA}{y} , \frac{x^{2} − A^{2}}{y} , \frac{x^{2} + A^{2}}{y} ).  (2)

For a positive integer A, a triple (a, b, c) of non-zero rational (integral) numbers such that a^{2} + b^{2} = c^{2} and A = |\frac{ab}{2} | is called a rational (integral) Pythagorean A-triple. Notice that if (a, b, c) is a rational Pythagorean A-triple, then A is a congruent number and |a|, |b|, |c| are the lengths of the sides of a right-angled triangle with area A. Notice also that we allow a, b, c to be negative. In particular, for any positive integers m and n with m > n, the triple

( \underbrace{2mn}_{a} , \underbrace{m^{2} − n^{2}}_{b} , \underbrace{m^{2} + n^{2}}_{c} )

is an integral Pythagorean A-triple. In this case, we obtain A = mn(m^{2} − n^{2}) and by (1)

(a, b, c) ↦ (\underbrace{m^{2}(m^{2} − n^{2})}_{x} , \underbrace{m^{2}(m^{2} − n^{2})^{2}}_{y}) .  (3)

Notice that in this case, the point (x, y) on C_{A} that corresponds to the integral Pythagorean A-triple (a, b, c) is an integral point.

Concerning the equations

m = n^{2} + nl + l^{2}  eq(m)

and

m^{2} = n^{2} + nl + l^{2}  eq(m^{2})

we first prove the following result (for a geometric representation of integral solutions of x^{2} + xy + y^{2} = m^{2}, see Halbeisen and Hungerbühler [5]):

Proposition 1. Let p_{1} < p_{2} < ⋯ < p_{j} be primes, such that p_{i} ≡ 1 (mod 6) for 1 ≤ i ≤ j, and let

m = ∏_{i=1}^{j} p_{i}.

(a) The number of positive, integral solutions (n, l) of eq(m) with l < n is 2^{j}.

(b) For each integral solution of eq(m), n and l are relatively prime and neither n nor l is a multiple of p_{i} (for 1 ≤ i ≤ j).

(c) The number of positive, integral solutions (n, l) of eq(m^{2}) with l < n is \frac{3^{j}−1}{2}.

(d) Among the \frac{3^{j}−1}{2} integral solutions solutions (n, l) of eq(m^{2}) with l < n we find 2^{j−1} solutions (n, l) such that n and l are relatively prime. In particular, if j = 1 and p ≡ 1 (mod 6), then the solution in positive integers l < n of

p^{2} = n^{2} + nl + l^{2},

such that n and l are relatively prime, is unique.

Proof.

(a) By Dickson [1, Exercises XXII.2, p. 80], the number of integral solutions of eq(m) is 6E(m), where E(m) is the excess of the number of divisors 3h + 1 of m over the number of divisors of the form 3h+2. By definition of m, E(m) = 2^{j}. Now, with each positive, integral solution (n, l) of eq(m) with 0 < l < n we obtain the following 12 pairwise different integral solutions:

(n, l), (l, n), (−n, −l), (−l, −n),
(−n, n + l), (n + l, −n), (n, −n − l), (−n − l, n),
(−l, n + l), (n + l, −l), (l, −n − l), (−n − l, l).

So, if e(m) denotes the number of positive, integral solutions (n, l) of eq(m) with l < n, then 6E(m) ≥ 12e(m). On the other hand, every integral solution of m = n^{2} + nl + l^{2} corresponds to a unique positive, integral solution (n, l) with l < n, which implies that 6E(m) = 12e(m) and consequently we obtain e(m) = 2^{j−1}.

(b) This follows immediately from the definition of m.

(c) Again by Dickson [1, Exercises XXII.2, p. 80], the number of integral solutions of eq(m^{2}) is 6E(m^{2}), where E(m^{2}) = 3^{j}. Let e(m^{2}) denote the number of positive, integral solutions (n, l) of m^{2} = n^{2} +nl+l^{2} with l < n. In addition to the 12e(m^{2}) integral solutions of eq(m^{2}), we have the 6 solutions

(m, 0), (0, m), (−m, 0), (0, −m), (−m, m), (m, −m).

So, 6E(m) = 12e(m) + 6 and consequently we obtain e(m) = \frac{3^{j}−1}{2}.

(d) For the sake of simplicity, let us call a positive, integral solution (n, l) of eq(m^{2}) with l < n a normal solution. Among the normal solutions (n, l) of eq(m^{2}), we distinguish between the ones with n and l relatively prime, which we call primitive solutions, and the other ones, which we call composite. For a given m, there is a one-to-one correspondence between composite solutions of eq(m^{2}) with (n, l) = d > 1 and primitive solutions of eq((m/d)^{2}) via the correspondence

m^{2} = n^{2} + nl + l^{2} ⟺ (m/d)^{2} = (n/d)^{2} + (n/d)(l/d) + (l/d)^{2} .

The proof is now by induction on j. For j = 1, the statement follows from the definition of m. Suppose the statement holds when m has j − 1 or less prime factors. Then, let m have j prime factors. We observe by the above correspondence that the number of composite solutions to eq(m^{2}) is

∑_{\begin{matrix}r|m \\ 0<r<m \end{matrix}} P (r^{2})

where P (r^{2}) is the number of primitive solutions to r^{2} = n^{2} + nl + l^{2}. By the inductive step, if r is comprised of i prime factors of m then P (r^{2}) = 2^{i−1}. Furthermore, the number of distinct factors r of m with i prime factors is \binom{j}{i}, and therefore, by the binomial theorem, the number of composite solutions to eq(m^{2}) is

∑_{\begin{matrix}r|m \\ 0<r<m \end{matrix}} P (r^{2}) = ∑_{i=1}^{j−1} \binom{j}{i}2^{i−1} = \frac{1}{2}((1 + 2)^{j} − 2^{j} − 1) = \frac{3^{j} − 1}{2} − 2^{j−1} .

Since the total number of solutions to eq(m^{2}) is \frac{3^{j}−1}{2}, by subtracting the number of composite solutions we finally obtain P (m^{2}) = 2^{j−1}, which completes the proof.

Let us now consider the relationship between positive, integral solutions of m^{2} = n^{2} + nl + l^{2} and integral Pythagorean triples.

If m, n, l are positive integers such that m^{2} = n^{2} + nl + l^{2}, then, for k := n + l, each of the following three triples

( \underbrace{2mn}_{a_{1}} , \underbrace{m^{2} − n^{2}}_{b_{1}} , \underbrace{m^{2} + n^{2}}_{c_{1}} ) ,

( \underbrace{2ml}_{a_{2}} , \underbrace{m^{2} − l^{2}}_{b_{2}} , \underbrace{m^{2} + l^{2}}_{c_{2}} ) ,

( \underbrace{2mk}_{a_{3}} , \underbrace{k^{2} − m^{2}}_{b_{3}} , \underbrace{k^{2} + m^{2}}_{c_{3}} ) ,

is an integral Pythagorean A-triple for

A = mn(m^{2} − n^{2}) = ml(m^{2} − l^{2}) = km(k^{2} − m^{2}) = klmn

(see Hungerbühler [6]). In particular, with m, n, l and (3) we obtain three distinct integral points on C_{A}.

As a matter of fact we would like to mention that the three integral points on C_{A} that correspond to an integral solution of m^{2} = n^{2} + nl + l^{2} lie on a straight line.

Let us now turn back to the curve C_{A}, where A is a congruent number. One can readily check that the three points (0, 0) and (±A, 0) are the only points on C_{A} of order 2 with respect to the group law of elliptic curves. Moreover, one can show that these three points, together with the point at infinity, are the only points of finite order (for an elementary proof of this result, which is based on a theorem of Fermat; see Halbeisen and Hungerbühler [4]). This implies that if A is a congruent number and (x_{0}, y_{0}) is a rational point on C_{A} with y_{0} ≠ 0, then the order of (x_{0}, y_{0}) is infinite. So, Mordell’s theorem (which states that the group of rational points on C_{A} is finitely generated) and the fundamental theorem of finitely generated abelian groups imply that the group of rational points on a congruent number curve C_{A} is isomorphic to

\underbrace{ℤ/2ℤ × ℤ/2ℤ}_{torsion group} × ℤ^{r},

where r > 0 is the aforementioned rank of C_{A}.

2 Rank at least two

Based on integral solutions of m^{2} = n^{2} + nl + l^{2}, we will show that there are infinitely many congruent number curves C_{A} with rank at least two — where many of the curves C_{A} have rank three or four and several curves have even rank five (see Section 3).

The following result, which can be found in Silverman and Tate [9, Chapter III.6.], allows us to compute the rank — or at least a lower bound of the rank — of certain elliptic curves. For simplicity, we state the result just for congruent number curves.

Proposition 2. Let b be a non-zero integer and let b̄:= −4b. Furthermore, let

B := {b_{1} ∈ ℤ : b_{1} | b, and b_{1} is square-free}

and

B̄ := {b̄_{1} ∈ ℤ : b̄_{1} | b̄, and b̄_{1} is square-free}.

Finally, let β_{b} and βb̄ be the number of integers b_{1} ∈ B and b̄_{1} ∈ B̄, respectively, such that the corresponding equation

b_{1}M^{4} + \frac{b}{b_{1}} e^{4} = N^{2}  (4)
b̄_{1}M̄^{4} + \frac{b̄}{b̄_{1}} ē^{4} = N̄^{2}  (5)

has integral solutions, where e ≠ 0, ē ≠ 0, and (M, e) = (M̄, ē) = 1.

Then the rank r of the curve y^{2} = x^{3} + bx satisfies the equation

2^{r} = \frac{β_{b} · βb̄}{4}.

Moreover, if (x, y) is a rational point on the curve y^{2} = x^{3} + bx with y ≠ 0, then one can write that point in the form

x = \frac{b_{1}M^{2}}{e^{2}} ,  y = \frac{b_{1}MN}{e^{3}} ,  (6)

where M, e, N is an integral solution of (4) with N > 0 and (M, e) = 1, and vice versa. The analogous statement holds for rational points on the curve y^{2} = x^{3} + b̄x with respect to equations of the form (5).

Now we are ready to prove

Theorem 3. Let m, n, l be pairwise relatively prime positive integers, where m = ∏_{i=1}^{j} p_{i} is a product of pairwise distinct primes p_{i} ≡ 1 (mod 6) and m^{2} = n^{2} + nl + l^{2}. Furthermore, let k := n + l and let A := klmn. Then the rank of the congruent number curve

C_{A} : y^{2} = x^{3} − A^{2}x

is at least two.

Proof. Since we have at least one rational point (x, y) on C_{A} with y ≠ 0, we know that the rank r of C_{A} is positive. So, to show that the rank of the curve C_{A} is at least two, it would be enough to show that β_{−A^{2}} ≥ 9. For this, we have to show that there are integral solutions for (4) for at least 9 distinct square-free integers b_{1} dividing −A^{2}, or equivalently, we have to find at least 9 rational points on C_{A}, such that the 9 corresponding integers b_{1} are pairwise distinct. Even though it would be enough to find integral solutions for (4) for at least 9 distinct square-free integers b_{1}, we shall give 16 solutions, such that a single additional solution for (4) would give us a rank of at least three (see Proposition 5).

Notice that because of (6), to compute b_{1} from a rational point P = (x, y) on C_{A} with x ≠ 0, it is enough to know the x-coordinate of P and then compute x mod ℚ^{∗} ^{2} (i.e., we compute x modulo squares of rationals). The x-coordinates of the three integral points we get by (1) from the three integral Pythagorean A-triples (a_{1}, b_{1}, c_{1}), (a_{2}, b_{2}, c_{2}), (a_{3}, b_{3}, c_{3}) generated by m, n, l, k, are

x_{1} = m^{2}(m^{2} − n^{2}) = m^{2}kl , x_{2} = m^{2}(m^{2} − l^{2}) = m^{2}kn , x_{3} = k^{2}(k^{2} − m^{2}) = k^{2}nl ,

and modulo squares, this gives us three values for b_{1} modulo squares, namely

b_{1,1} ≡ kl , b_{1,2} ≡ kn , b_{1,3} ≡ nl .

Now, exchanging in each of the three integral Pythagorean A-triples the two catheti a_{i} and b_{i} (for i = 1, 2, 3), we obtain again three distinct integral points on C_{A}, whose x-coordinates gives us again three values for b_{1} modulo squares, namely

b_{1,4} ≡ mn , b_{1,5} ≡ ml , b_{1,6} ≡ mk .

Finally, if we replace each hypotenuse c_{j} of these six integral Pythagorean A-triples with −c_{j}, we obtain again six distinct integral points on C_{A}, whose x-coordinates give us six values for b_{1} modulo squares, namely

\begin{aligned}b_{1,7} & ≡ & −kl , & b_{1,8} & ≡ & −kn , & b_{1,9} & ≡ & −nl \\ b_{1,10} & ≡ & −mn , & b_{1,11} & ≡ & −ml , & b_{1,12} & ≡ & −mk . \end{aligned}

In addition to these 12 integral points on C_{A} (and the corresponding b_{1}’s), we have the two integral points (±A, 0) on C_{A}, which give us two more values for b_{1} modulo squares, namely

b_{1,13} ≡ klmn and b_{1,14} ≡ −klmn.

Now, it remains to show that the square-free parts of the b_{1,j}’s are pairwise distinct. By assumption, m is square-free and k, l, n are pairwise relatively prime. Therefore, if for some i, j with 1 ≤ i < j ≤ 14, b_{1,i} ≡ b_{1,j} (mod ℚ^{∗} ^{2}), at least two of the integers k, l, n are squares, say n = u^{2}, and l = v^{2} or k = v^{2}. Then

m^{2} = u^{4} + u^{2}v^{2} + v^{4} (in the case when l = v^{2}),

or

m^{2} = u^{4} − u^{2}v^{2} + v^{4} (in the case when k = v^{2}).

If l = v^{2}, this implies that u^{2} = 1 and v = 0, or u = 0 and v^{2} = 1, and if k = v^{2}, this implies that u^{2} = 1 and v = 0, u = 0 and v^{2} = 1, or u^{2} = v^{2} = 1 (see, for example, Mordell [8, p. 19f ] or Euler [3, p. 16]). So, at most one of the integers k, l, n is a square, which shows that at least 14 equations of the form (4) — for different square-free integers b_{1} dividing A^{2} — have integral solutions. Notice that so far, because the corresponding points on C_{A} are integral, we always had |e| = 1.

Now, we show that there are also solutions for (4) with b_{1,15} = 1 and b_{1,16} = −1. Assume first that there is a solution for (4) with b_{1,15} = 1 and e = 1, i.e., there are positive integers M and N such that

M^{4} − A^{2} = N^{2}.

Then we also have

A^{2}M^{4} − A^{4} = (AN)^{2},

which shows that M̃ := A, ẽ:= M, and Ñ := AN, satisfy

−M̃^{4} + A^{2}ẽ^{4} = Ñ^{2},

and hence, there is a solution for (4) with b_{1,16} = −1. Notice that since |ẽ| ≠ 1, the corresponding point on C_{A} is not an integral point.

It remains to find a solution for (4) with b_{1,15} = 1 and e = 1. Since m, n, l, k are pairwise relatively prime positive integers and k = n + l, exactly one of n, l, k is even, i.e., at least one of n and l is odd. Without loss of generality, assume that n is odd. Furthermore, by definition of m, m is odd. Let p := \frac{m+n}{2} and q := \frac{m−n}{2}. Then p and q are positive integers. Now, m = p + q, n = p − q, and m^{2} − n^{2} = 4pq, and since m^{2} − n^{2} = kl, we have

A = klmn = 4pq(p + q)(p − q) .

Notice that since pq(p + q) is even, we have A ≡ 0 (mod 8). An easy calculation shows that M := p^{2} + q^{2} and N := (p^{2} − q^{2})^{2} − (2pq)^{2} satisfy

M^{4} − A^{2} = N^{2}.

So, there is a solution for (4) with b_{1,15} = 1 and e = 1, which gives us again an integral point on C_{A}

This shows that β_{−A^{2}} ≥ 16 and completes the proof.

As an immediate consequence we get the following

Corollary 4. Let m, n, l be as in Theorem 3 and let q be a non-zero integer. Then the rank of the curve C_{Aq^{4}} is at least two.

Proof. Notice that if m, n, l are such that m^{2} = n^{2} + nl + l^{2}, then, for mq, nq, lq, we have (mq)^{2} = (nq)^{2} + nq · lq + (lq)^{2}, which implies that for Ã = kq · lq · mq · nq = Aq^{4}, the rank of the curve CÃ is at least two.

3 Rank at least three

Proposition 5. For A = 341 880, the rank of the curve C_{A} is at least three.

Proof. For k = 40, l = 7, m = 37, n = 33, we have A = klmn, m^{2} = n^{2} + nl + l^{2}, and k = n+l. Thus, by Theorem 3, the rank of the curve C_{A} is a least two. Now, for b_{1,17} = −30, which is distinct from the square-free values of b_{1,1}, …, b_{1,16}, we get that M = 98, e = 1, N = 33 600 is an integral solution of (4), which implies that the rank of C_{A} is at least three. In fact, with the help of one can show that the rank of C_{A} is equal to three.

As a final remark concerning the rank of congruent number curves, we would like to mention that with the help of we found that many of the curves which correspond to an integral solution of m^{2} = n^{2} + nl + l^{2} have rank 3 or higher. In fact, we found plenty of curves of rank 3 or 4, as well as the following curves of rank 5:

\begin{matrix}A = klmn & m = ∏ p_{i} & l & n & k = n + l \\ 237 195 512 400 & 7 · 127 & 464 & 561 & 1 025 \\ 8 813 542 297 560 & 7 · 13 · 37 & 232 & 3 245 & 3 477 \\ 10 280 171 942 040 & 37 · 67 & 741 & 2 024 & 2 765 \\ 81 096 660 783 600 & 37 · 103 & 2 139 & 2 261 & 4 400 \\ 225 722 120 463 840 & 13 · 19 · 31 & 505 & 7 392 & 7 897 \\ 457 485 316 904 280 & 7 · 31 · 37 & 895 & 7 544 & 8 439 \\ 5 117 352 889 729 080 & 67 · 223 & 1 551 & 14 105 & 15 656 \\ 281 692 457 452 791 000 & 79 · 409 & 9 064 & 26 811 & 35 875 \\ 24 666 188 870 481 576 600 & 13 · 31 · 223 & 46 169 & 57 400 & 103 569 \end{matrix}

Of special interest might be values of A which are related to an m with few factors, especially to primes m. Recall that if m is prime, then the integers n, l, n + l such that m^{2} = n^{2} + nl + l^{2} are unique, and therefore, A(m) := (n+l)lmn is determined by m. Among the 666 prime numbers m ≤ 11 113 with m ≡ 1 (mod 6), we found the following 30 values of m such that C_{A(m)} has rank 4:

127, 139, 181, 277, 337, 709, 769, 823, 829, 883, 1051, 1087, 1213,
1747, 1777, 1873, 2137, 2287, 2377, 2467, 2521, 3529, 3877, 3931,
4129, 4999, 5521, 7573, 9601, 10711.

However, we did not find any prime m ≡ 1 (mod 6) such that C_{A(m)} has rank 5.

The preceding observations might indicate that the congruent number curves C_{A} constructed in Theorem 3 are candidates for high rank congruent number elliptic curves (for another approach, see Dujella, Janfada, Salami [2]).

4 Acknowledgment

We would like to thank the referee for his or her valuable remarks which greatly helped to improve this article.

References

[1] L. E. Dickson, Introduction to the Theory of Numbers, The University of Chicago Press, 7th edition, 1951.

[2] A. Dujella, A. S. Janfada, and S. Salami, A search for high rank congruent number elliptic curves, J. Integer Seq. 12 (2009), Article 09.5.8.

[3] L. Euler, De binis formulis speciei xx + myy et xx + nyy inter se concordibus et discordibus (Conventui exhibuit die 5. Junii 1780), Mémoires de l’Académie impériale des sciences de St. Pétersbourg, 5e série, Tome VIII (1817–18), 3–45.

[4] L. Halbeisen and N. Hungerbühler, A theorem of Fermat on congruent number curves, Hardy-Ramanujan J. 41 (2018), 15–21.

[5] L. Halbeisen and N. Hungerbühler, A geometric representation of integral solutions of x^{2} + xy + y^{2} = m^{2}, Quaest. Math., to appear.

[6] N. Hungerbühler, A proof of a conjecture of Lewis Carroll, Math. Mag. 69 (1996), 182–184.

[7] J. A. Johnstone and B. K. Spearman, Congruent number elliptic curves with rank at least three, Canad. Math. Bull. 53 (2010), 661–666.

[8] L. J. Mordell, Diophantine Equations, Academic Press,1969.

[9] J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer, 2nd edition, 2015.

[10] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. Published electroni-

cally at https://oeis.org, 2019.

2010 Mathematics Subject Classification: Primary 11G05; Secondary 11D09. Keywords: congruent number elliptic curves, Pythagorean triples.

(Concerned with sequence \underline{A003273}.)

Received October 29 2018; revised versions received January 25 2019; February 19 2019; May 15 2019; May 16 2019. Published in Journal of Integer Sequences, May 16 2019.

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