ab + 1 = r^{2}, ac + 1 = s^{2}, bc + 1 = t^{2}.
In order to extend this triple to a quadruple, we have to solve the system
 (6.1) 
It is natural idea to assign to this system the elliptic curve
 (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 pointsP = (0, 1), S = (1/abc, rst/abc).
It is not so obvious, but it is easy to verify that S = 2R, whereR = ((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 xcoordinate of the point
T ∈
E(Q) satisfies
(6.1) if and only if 
By Theorem 6.1 and the relation
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 xcoordinate of the point
T ∈
E(Q) satisfies
(6.1), then for the points

Corollary 6.1: Every Diophantine quadruple

In [62], it was shown that if the number e in Corollary 6.1
is obtained by construction from Theorem 6.2, then
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
In general, we may expect that the points P and S
are two independent points of infinite orders, and therefore that
E : y^{2} = (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 curvesE_{k}: y^{2} = ((k  1)x + 1) ((k + 1)x + 1)(4kx + 1),
under assumption thatAnalogous result was proved in [97] for the family
y^{2} = (F_{2k}x + 1) (F_{2k+2}x + 1) (F_{2k+4}x + 1)
under assumption that rank is equal to 1, and in [186] for the families
y^{2} = ((k  1)x + 1)
((k + 1)x + 1)((16k^{3}  4k)x + 1),
y^{2} = ((k  1)x + 1)
((k + 1)x + 1)((64k^{5}  48k^{3} + 8k)x + 1)
In [91], Dujella & Pethő considered the family
C_{k}: y^{2}=(x + 1)(3x + 1) (c_{k}x + 1),
where the sequence (c_{k}) is given byc_{1} = 8, c_{2} = 120, c_{k+2} = 14c_{k+1}  c_{k} + 8,
i.e. {1, 3, c_{k}} is a Diophantine triple. They proved that ifx ∈ {1, 0, c_{k1}, c_{k+1}}.
Jacobson & Williams [112] showed that the same result is unconditionally true for all
In [81], Herrmann, Pethő & Zimmer
computed all Sintegral points on some elliptic curves
associated with the D(256)quintuple
In the proof, they considered the elliptic curve E' given by
E' : y^{2} = x^{3}  9494304273243x + 11224300076670688758.
This curve has rank 4 and possesses an exceptionally large number of integral points. Namely,B(T) = sup { rank (E (Q)) : torsion group of E is T }.
In [87], an example was constructed
which shows that
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
y^{2} = (bx + 1)(d_{+}x + 1)(d_x + 1)
has at least four independent points of infinite order, namely, points with xcoordinates0, a, c, 1 / (bd_{+}d_).
In [87], oneparametric solution of the equation d_{+}d_ + 1 = w^{2} was found. In that way, an elliptic curve over Q(t) with rankThis 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
y^{2} = (ax + 1)(bx + 1)(cx + 1)(dx + 1).
By the substitutiont = y(d  a)(d  b)(d  c) / (dx + 1)^{2}, s = (ax + 1)(d  b)(d  c) / (dx + 1),
we obtain the following elliptic curveE*: t^{2} = s(s + (b  a)(d  c))(s + (c  a)(d  b)).
We have three nontrivial rational 2torsion 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) ).
(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 iff2P = ± Q;
The rational Diophantine quintuple{a, b, c, d, e} is regular iffR ± P = ± Q.
These characterizations suggest that if an irregular
rational Diophantine quadruple
{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 groupIt is possible to construct rational Diophantine triples {a, b, c} such that the elliptic curve
y^{2} = (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 thatFurthermore, the triples
{1270 / 2323, 5916 / 2323, 664593861324 / 12535672267},
{3164 / 491, 10692 / 491, 302996685420 / 118370771}
induce the elliptic curves with the torsion group{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
They showed that the curve with the torsion group
{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:The information about current rank records for all torsion groups may be found in this table.
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}
1. Introduction
2. Diophantine quintuple conjecture
3. Sets with the property D(n)
4. Connections with Fibonacci numbers
5. Rational Diophantine mtuples
7. Various generalizations
8. References
Diophantine mtuples page  Andrej Dujella home page 