6. Connections with elliptic curves


6.1. Diophantine triples and elliptic curves

Let {a, b, c} be a (rational) Diophantine triple, i.e.

ab + 1 = r2,     ac + 1 = s2,     bc + 1 = t2.

In order to extend this triple to a quadruple, we have to solve the system

ax + 1 =  □,       bx + 1 =  □,       cx + 1 =  □.
(6.1)

It is natural idea to assign to this system the elliptic curve

E :       y2 = (ax + 1)(bx + 1)(cx + 1).
(6.2)

We say that the elliptic curve E is induced by the Diophantine triple {a, b, c}.

There are three rational points on E of order 2:

A = (-1/a, 0),       B = (-1/b, 0),       C = (-1/c, 0),

and also obvious rational points

P = (0, 1),       S = (1/abc, rst/abc).

It is not so obvious, but it is easy to verify that S = 2R, where

R = ((rs + rt + st + 1) / abc, (r + s)(r + t)(s + t) / abc).

It is clear that every rational solution of the original system (6.1) induce a rational point on E. The question is which rational points on E induce a rational solution of (6.1). The answer is given in the following theorem (see [97] and [37, Anwendung 1]).

Theorem 6.1: The x-coordinate of the point T ∈ E(Q) satisfies (6.1) if and only if T − P ∈ 2E(Q).

By Theorem 6.1 and the relation S = 2R, it follows that the numbers x(P + S) and x(P - S) satisfy the system (6.1). It is easy to check that

x(P + S) = d_,         x(P - S) = d+,

where d_ and d+ are defined in Chapter 2.

The addition and subtraction of the point S has another interesting property (see [62, 97]):

Theorem 6.2: If x-coordinate of the point T ∈ E(Q) satisfies (6.1), then for the points T ± S = (u,v) it holds that xu + 1 is a square.

Corollary 6.1: Every Diophantine quadruple {a, b, c, d} can be extended to a rational Diophantine quintuple {a, b, c, d, e}.

In [62], it was shown that if the number e in Corollary 6.1 is obtained by construction from Theorem 6.2, then e < 1, and therefore e is not a positive integer.

The next question is what can be said about torsion group and rank of E. In [97, 268], it was proved that if a, b, c are positive integers, then E(Q)tors = Z/2Z × Z/2Z or Z/2Z × Z/6Z, and rank E(Q) ≥ 1.

In general, we may expect that the points P and S are two independent points of infinite orders, and therefore that rank E (Q) ≥ 2. However, if c is the smallest possible (greater than a and b), i.e. c = a + b + 2r, then we have S = -2P.

6.2. Integer points on elliptic curves induced by Diophantine triples

Let us consider now the problem of finding all integer points on the elliptic curve

E :       y2 = (ax + 1)(bx + 1)(cx + 1)

We have always the following integer points:

(0, ±1),     (d+, ±(at + rs)(bs + rt) (cr + st)),     (d_, ±(at - rs)(bs - rt)(cr - st)),

and also (-1,0) if 1 ∈ {a, b, c}. For some families of Diophantine triples it is possible to prove that there are no other integer points on E. This was proved in [89] for elliptic curves

Ek:       y2 = ((k - 1)x + 1) ((k + 1)x + 1)(4kx + 1),

under assumption that rank Ek (Q) = 1. The same result was proved also for two subfamilies of rank 2 and one subfamily of rank 3. The statement is also verified for all k such that 3 ≤ k ≤ 1000 (and extended to k ≤ 5000 in [184]). The condition rank Ek (Q) = 1 is not unrealistic since the generic rank, rank E (Q(k)), is equal to 1. In the range 2 ≤ k ≤ 100 we have the following distribution of ranks (found using MWRANK and SIMATH): 41 cases of rank 1, 49 cases of rank 2 and 9 cases of rank 3.

Analogous result was proved in [97] for the family

y2 = (F2kx + 1) (F2k+2x + 1) (F2k+4x + 1)

under assumption that rank is equal to 1, and in [186] for the families

y2 = ((k - 1)x + 1) ((k + 1)x + 1)((16k3 - 4k)x + 1),
y2 = ((k - 1)x + 1) ((k + 1)x + 1)((64k5 - 48k3 + 8k)x + 1)

under assumption that rank is equal to 2.

In [91], Dujella & Pethő considered the family

Ck:       y2=(x + 1)(3x + 1) (ckx + 1),

where the sequence (ck) is given by

c1 = 8,   c2 = 120,   ck+2 = 14ck+1 - ck + 8,

i.e. {1, 3, ck} is a Diophantine triple. They proved that if rank Ck (Q) = 2 or k ≤ 40, with the possible exceptions of k = 23 or 37, then all the integer points on Ck are given by

x ∈ {-1, 0, ck-1, ck+1}.

Jacobson & Williams [112] showed that the same result is unconditionally true for all k ≤ 100, with the possible exception of k = 37, for which the result holds under the assumption of the Extended Riemann Hypothesis (see also [147]).

In [81], Herrmann, Pethő & Zimmer computed all S-integral points on some elliptic curves associated with the D(256)-quintuple {1, 33, 105, 320, 18240}. They proved that if x and z are integers such that gcd(x,z) = 1, z is at most divisible by the primes from the set S = {2, 3, 5, 11, 19, 29, 1621}, and x + z2, 33x + z2, 320x + z2 and 18240x + z2 are perfect squares, then (x,|z|) ∈ {(0, 1), (105, 24), (117, 2 · 29), (42315, 1621)}.

In the proof, they considered the elliptic curve E' given by

E' :       y2 = x3 - 9494304273243x + 11224300076670688758.

This curve has rank 4 and possesses an exceptionally large number of integral points. Namely, |E(Z)| = 208.

6.3. High-rank elliptic curves

Let T be an admissible torsion group for an elliptic curve over the rationals. Define

B(T) = sup { rank (E (Q)) : torsion group of E is T }.

In [87], an example was constructed which shows that B (Z/2Z × Z/2Z) ≥ 7.

The construction starts with a rational Diophantine triple {a, b, c}. Assume that d+d_ + 1 is a perfect square. Then we may expect that the elliptic curve

y2 = (bx + 1)(d+x + 1)(d_x + 1)

has at least four independent points of infinite order, namely, points with x-coordinates

0,     a,     c,     1 / (bd+d_).

In [87], one-parametric solution of the equation d+d_ + 1 = w2 was found. In that way, an elliptic curve over Q(t) with rank ≥ 4 was obtained. By specialization of parameter t, using MWRANK, a curve with rank = 7 is found.

This result was improved in [90] using different construction. Let {a, b, c, d} be a rational Diophantine quadruple. We may assign to this quadruple the elliptic curve

y2 = (ax + 1)(bx + 1)(cx + 1)(dx + 1).

By the substitution

t = y(d - a)(d - b)(d - c) / (dx + 1)2,       s = (ax + 1)(d - b)(d - c) / (dx + 1),

we obtain the following elliptic curve

E*:       t2 = s(s + (b - a)(d - c))(s + (c - a)(d - b)).

We have three non-trivial rational 2-torsion points on E*, and another two obvious rational points:

P = ( (b - a)(c - a), (b - a)(c - a)(d - a) ),
Q = ( (ad + 1)(bc + 1), √(ab + 1)(ac + 1) (ad + 1)(bc + 1)(bd + 1)(cd + 1) ).

Assume that the Diophantine quadruple {a, b, c, d} can be extended to the Diophantine quintuple {a, b, c, d, e}. Then we have on E* another rational point R with x-coordinate

(de + 1)(b - a)(c - a) / (ae + 1).

The following characterizations of the notions of regular Diophantine quadruples and quintuples are given in [90]:

The rational Diophantine quadruple {a, b, c, d} is regular iff 2P = ± Q;
The rational Diophantine quintuple {a, b, c, d, e} is regular iff R ± P = ± Q.

These characterizations suggest that if an irregular rational Diophantine quadruple {a, b, c, d} is contained in an irregular rational Diophantine quintuple, then we may expect that the rank of the corresponding elliptic curve E* over Q is ≥ 3. In [90], this idea was applied on quadruples which are subsets of Gibbs' examples of Diophantine sextuples from [144, 80] and several elliptic curves were obtained with rank equal to 8, corresponding to the quadruples

{17 / 448, 2145 / 448, 23460 / 7, 2352 / 7921},

{32 / 91, 60 / 91, 1878240 / 1324801, 15345900 / 12215287},

{252 / 115, 559 / 1380, 24264935 / 2979076, 16454108 / 1703535},

{81 / 1400, 2875 / 168, 4928 / 3, 5696 / 4725},

{805 / 1404, 1105 / 108, 21280 / 351, 41067 / 128164},

{12 / 119, 620 / 357, 39984 / 18769, 1125176416 / 1493609907},

{559 / 1380, 252 / 115, 24364935 / 2979076, 16454108 / 1703535}.

Let us mention that these results were improved, with different methods, by Kulesz & Stahlke, Dujella & Kulesz, Eroshin and Aguirre, Dujella & Peral. They found several examples of elliptic curves with torsion group Z/2Z × Z/2Z and rank equal to 10, 11 or 12, while Elkies found an example which shows that B(Z/2Z × Z/2Z) ≥ 15.

It is possible to construct rational Diophantine triples {a, b, c} such that the elliptic curve

y2 = (ax + 1)(bx + 1)(cx + 1)

has larger torsion group and relatively high rank. Let us mention three examples (see [149]):

{-22552 / 5129, 5129 / 22552, -52463190 / 14458651},

{39123 / 96976, 12947200 / 418209, 42427 / 1104},

{145 / 408, -408 / 145, -145439 / 59160},

which show that B(Z/2Z × Z/4Z) ≥ 7, B(Z/2Z × Z/6Z) ≥ 4 and B(Z/2Z × Z/8Z) ≥ 3, respectively. Connections of Diophantine triples and elliptic curves with torsion group Z/2Z × Z/8Z are also discussed in [171]. It is interesting that any elliptic curve with torsion group Z/2Z × Z/8Z over Q can be obtained as a curve induced by a rational Diophantine triple.

Furthermore, the triples

{1270 / 2323, 5916 / 2323, 664593861324 / 12535672267},

{3164 / 491, 10692 / 491, 302996685420 / 118370771}

induce the elliptic curves with the torsion group Z/2Z × Z/2Z and the rank equal to 9 (see [149]), the triple

{1235 / 69, 5146848 / 69277, 91544915 / 19120452}

gives a curve with rank equal to 10, while the triple

{795025 / 3128544, -22247424 / 7791245, 24807390285149 / 97501011189120}

gives a curve with rank equal to 11 (Aguirre, Dujella & Peral [209]). Other examples with rank equal to 11 and a curve over Q(t) with rank equal to 6 were constructed in [378]. An example with rank equal to 12, corresponding to the triple

{6125241375 / 11907531272, 5535371271425 / 14277129995128, −273138178560 / 153430695649},

is constructed by Dujella & Peral in [438]. In 2023, they found another example with rank 12, corresponding to the triple

{-677586092432400 / 7491447777801647, 5414607463683456 / 2961735167968093, 151762875459723 / 43938110133734}.

Dujella & Peral [263] found the triple

{301273 / 556614, -556614 / 301273, -535707232 / 290125899}

which induces the elliptic curve with the torsion group Z/2Z × Z/4Z and the rank equal to 9, what is the current record for this torsion group.

They showed that the curve with the torsion group Z/2Z × Z/6Z and the rank equal to 6 (which is the current record for this torsion group), originally found by Elkies, can also be obtained as the curve induced by the rational Diophantine triple

{31269599 / 31628160, -23721120 / 31269599, 1461969791 / 7144352640}.

Dujella, Jukic Bokun & Soldo [336] studied possible torsion groups of elliptic curves induced by Diophantine triples over quadratic fields and showed that apart from the four group which are possible over Q, over quadratic fields three additional group appear: Z/2Z × Z/10Z, Z/2Z × Z/12Z and Z/4Z × Z/4Z.

The information about current rank records for all torsion groups may be found in this table.

6.4. Curves with rank 0

It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. E.g. the triples {-1/2, 3/2, 2} and {-1/3, 8/3, 3} induce elliptic curves with torsion group Z/2Z × Z/4Z and rank 0, the triples {-1/12, 21/4, 20/3} and {-15/28, 12/7, 7/4} induce elliptic curves with torsion group Z/2Z × Z/6Z and rank 0, while the triple {-3/4, 7/12, 4/3} induces elliptic curve with torsion group Z/2Z × Z/8Z and rank 0 (see [306]).

It is much harder to find such examples of rational Diophantine triples with positive elements. It seems that only such known example is the triple 

{69705492951192675600645567228019184577147632882703132983 / 132014843349912467692901303836561266921302184459536763120, 

 47826829880079829075801189563942620732062701095548790400 / 122336669420709509303637442647966391336596694969835459327, 

47982111146649404421749331709393501777791774558546217987550257759801 / 15400090753918257364093484910580652390786084055043677020804056653840}
which induces elliptic curve with torsion group Z/2Z × Z/8Z and rank 0.

1. Introduction
2. Diophantine quintuple conjecture
3. Sets with the property D(n)
4. Connections with Fibonacci numbers
5. Rational Diophantine m-tuples
7. Various generalizations
8. References

Diophantine m-tuples page Andrej Dujella home page