Let {a, b, c, d} be a rational Diophantine quadruple, i.e.
ab + 1 = r_{1}^{2}, ac + 1 = r_{2}^{2}, ad + 1 = r_{3}^{2},
bc + 1 = r_{4}^{2}, bd + 1 = r_{5}^{2}, cd + 1 = r_{6}^{2}.
Assume that abcd ≠ 1 and definee = ((a+b+c+d)(abcd+1) + 2abc + 2abd + 2acd+2bcd ± 2r_{1}r_{2}r_{3}r_{4}r_{5} r_{6}) / (abcd  1)^{2}.
In [62] it was proved that ae + 1, be + 1, ce +1 and de +1 are perfect squares. E.g.ae + 1 = (ar_{4}r_{5}r_{6} ± r_{1}r_{2}r_{3})^{2} / (abcd  1)^{2}.
A rational Diophantine quintuple {a, b, c, d, e} obtained by this construction is called regular. For characterization of regular Diophantine quintuples in the terms of elliptic curves see Chapter 6.3.For example, if we apply this construction to Diophantus' original set {1/16, 33/16, 17/4, 105/16}, we obtain
e_{1} = 549120 / 10201, e_{2} = 26880 / 177241.
If abcd = 1, then ab, ac, ad, bc, bd, cd are all perfect squares (see [452]) and {a, b, c, d} can be extended to a rational Diophantine quintuple by
e = ((a+bcd)^{2}  4(ab+1)(cd+1)) / (4d(ab+1)(ac+1)(bc+1)).
In these examples all elements are positive. There exist Diophantine sextuples with mixed signs (see [187]). E.g.
{19/12, 33/4, 52/3, 60/2209, 495/24964, 595/12},
{31/84, 9/7, 49/12, 160/21, 455/3468, 7200/2023},
{7/40, 75/56, 41/70, 5376/4805, 300288/241115, 165/224},
{27/35, 35/36, 1007/1260, 352/315, 72765/106276, 5600/4489},
{8/17, 85/72, 763/1224, 18360/11449, 4914/8993, 332605/496008}.
Moreover, there are infinitely many rational Diophantine sextuples contaning the triple {15/14, 16/21, 7/6}. Some of them are
{15/14, 16/21, 7/6, 1680/3481, 910/1083, 624/847},
{15/14, 16/21, 7/6, 2925534221610/3772144724401, 269287270560/208966922641, 288918935136/401994662167},
{15/14, 16/21, 7/6, 201715143251025578282013360/191025652605458897388287161, 682169558073146933103328560/833143872544491452774840761, 28016255787642946981919458/45679215332619641097811101}.
The same result holds also for some other triples with mixed signs, e.g. {48/17, 35/102, 17/6}, {56/69, 165/184, 23/24}, but also to the triple {36534805866201747/2323780774755404, 1065197767305747/13609226201091404, 3802080647508196/6238332600753747} with positive elements.
In 2017, Dujella, Kazalicki, Mikic & Szikszai [323] proved that there exist infinitely many rational Diophantine sextuples parametrized by a rational parameter t:
a = 18t(t^{2}1) / ((t^{2}6t+1)(t^{2}+6t+1)),
b = (t1)(t^{2}+6t+1)^{2} / (6t(t+1)(t^{2}6t+1)),
c = (t+1)(t^{2}6t+1)^{2} / (6t(t1)(t^{2}+6t+1)),
d = d_{1} / d_{2}, where
d_{1} = 6(t+1)(t1)(t^{2}+6t+1)(t^{2}6t+1)(8t^{6}+27t^{5}+24t^{4}54t^{3}+24t^{2}+27t+8)(8t^{6}27t^{5}+24t^{4}+54t^{3}+24t^{2}
27t+8)(t^{8}+22t^{6}174t^{4}+22t^{2}+1),
d_{2} = t(37t^{12}885t^{10}+9735t^{8}13678t^{6}+9735t^{4}885t^{2}+37)^{2},
e = e_{1} / e_{2}, where
e_{1} =
2t(4t^{6}111t^{4}+18t^{2}+25)(3t^{7}+14t^{6}42t^{5}+30t^{4}+51t^{3}+18t^{2}12t+2)(3t^{7}14t^{6}42t^{5}
30t^{4}+51t^{3}18t^{2}12t2)(t^{2}+3t2)(t^{2}3t2)(2t^{2}+3t1)(2t^{2}3t1)(t^{2}+7)(7t^{2}+1),
e_{2} =
3(t+1)(t^{2}6t+1)(t1)(t^{2}+6t+1)(16t^{14}+141t^{12}1500t^{10}+7586t^{8}2724t^{6}+165t^{4}+424t^{2}12)^{2},
f = f_{1} / f_{2}, where
f_{1} =
2t(25t^{6}+18t^{4}111t^{2}+4)(2t^{7}12t^{6}+18t^{5}+51t^{4}+30t^{3}42t^{2}+14t+3)(2t^{7}+12t^{6}+18t^{5}
51t^{4}+30t^{3}+42t^{2}+14t3)(2t^{2}+3t1)(2t^{2}3t1)(t^{2}3t2)(t^{2}+3t2)(t^{2}+7)(7t^{2}+1),
f_{2} =
3(t+1)(t^{2}6t+1)(t1)(t^{2}+6t+1)(12t^{14}424t^{12}165t^{10}+2724t^{8}7586t^{6}+1500t^{4}141t^{2}16)^{2}.
Theorem 5.1: There exist infinitely many rational Diophantine sextuples. Moreover, there exist infinitely many rational Diophantine sextuples with positive elements, and also with any combination of signs. 
In fact, it was proved in [323] that there exist infinitely many rational Diophantine triples each of which can be extended to rational Diophantine sextuples on infinitely many ways.
In 2016, Piezas (see here and [341]) found a simpler parametric formula for rational Diophantine sextuples:
a = (t^{2}2t1)(t^{2}+2t+3)(3t^{2}2t+1) / (4t(t^{2}1)(t^{2}+2t1)),
b = 4t(t^{2}1)(t^{2}2t1) / ((t^{2}+2t1)^{3}),
c = 4t(t^{2}1)(t^{2}+2t1) / ((t^{2}2t1)^{3}),
d = (t^{2}+2t1)(t^{2}2t+3)(3t^{2}+2t+1) / (4t(t^{2}1)(t^{2}2t1)),
e = (t^{5}+14t^{3}t) / (t^{6}7t^{4}+7t^{2}1),
f = (3t^{6}13t^{4}+13t^{2}3) / (4t(t^{4}6t^{2}+1)).
In 2016, Gibbs [349] found examples of "almost" septuples, namely, rational Diophantine quintuples which can be extended to rational Diophantine sextuples on two different ways, so that only one condition is missing that these seven numbers form a rational Diophantine septuple:
{243/560, 1147/5040, 1100/63, 7820/567, 95/112} can be extended with 38269/6480 or 196/45;
{7657/420, 480/91, 441/260, 425/1092, 191840/273} can be extended with 13/105 or 43953/39605;
{518/45, 7344/185, 4004/1665, 25900/690561, 216/185} can be extended with 100/333 or 166600/37.
{8415/2366, 1575/6728, 256/105, 943/3360, 1085/4056} can be extended with 531/280 or 242515/80736;
{7119/9280, 1131/1280, 164/145, 1925/1856, 2100/24389} can be extended with 18444/22445 or 1532244/1584845;
{13/3, 2223/3136, 4048/1911, 4851/832, 7696/147} can be extended with 47/2496 or 23616/148837;
{629/3828, 108/319, 1276/243, 2457/1276, 15776/3993} can be extended with 13156/73167 or 102641/28188;
{15200/59643, 21413/13824, 27375/8192, 424/675, 3441/12800} can be extended with 46008/16129 or 3198/1225;
{154275/2376416, 1617/2720, 1950/1309, 2737/5280, 13144/19635} can be extended with 55352/37905 or 18176/19635;
{215061/322580, 15200/24843, 493/1140, 960/931, 9295/11172} can be extended with 23904/16055 or 31119/64220;
{2255/13524, 24480/12397, 14773/3381, 272320/195657, 207/44} can be extended with 589/3036 or 1047179/1606044;
{91/180, 3895/4032, 512/315, 2135/3844, 1053/2240} can be extended with 9855/5488 or 68425/46656;
{7084/765, 15903/2380, 833/180, 260/119, 1548/14875} can be extended with 10115/191844 or 109395/221788;
{8075/1452, 544/135, 475/204, 189/340, 10597664/64708935} can be extended with 37/255 or 1357/30855;
{119/864, 62/459, 77112/52441, 2849/1632, 697/96} can be extended with 232/51 or 20736/2873;
{14212/15435, 6768/665, 2392/5985, 2044/95, 270/133} can be extended with 146300/31329 or 26600/114921;
{3/70, 51987/280, 12103/4320, 2560/1701, 134589/1024000} can be extended with 3025715/864 or 105/8;
{19908/65, 6204/455, 2275/192, 1300/7581, 3536/5145} can be extended with 48351/29120 or 34397/87360.
{11825/2016, 51200/693, 9163/92160, 497/990} can be extended with {10989/280, 551/3080}, {10989/280, 19035/9856} or {551/3080, 17577/1760}.
Herrmann, Pethoe & Zimmer [81] proved in 1999 that a rational Diophantine quadruple has only finitely many extensions to a rational Diophantine quintuple. They proved that the conditions on the fifth element of the quintuple lead to a curve of genus 4, and then they applied Faltings' theorem.
Gibbs [349] found two rational Diophantine quadruples which can be
extended to quintuples in six different ways:
the quadruple {81/1400, 5696/4725, 2875/168, 4928/3}
can be extended to a quintuple using any one of these rationals:
98/27, 104/525, 96849/350, 1549429/1376646, 3714303488/6103383075, 7694337252154322/1857424629984075;
while the quadruple {152/357, 2665/2856, 3906/17, 1224/12943} can be extended with any of1519/408, 4505/168, 1959335/7824984, 13303605/1077512, 73026883629/17054089928, 515358540182255/7116911275416.
It follows easily from Theorem 3.2 that for every rational number q there exist infinitely many rational D(q)quadruples. This motivates the following question:
For which rational numbers q there exist infinitely many rational D(q)quintuples?
An affirmative answer to this question is known for rationals
of the forms
y^{2} = x^{3} + 42x^{2} + 432x + 1296
has positive rank, see [88]).In 2012, Dujella & Fuchs [238] obtained much more general result by proving that for infinitely many squarefree numbers q there are infinitely many rational D(q)quintuples, by considering twists of the elliptic curve
y^{2} = x^{3} + 86x^{2} + 825x
with positive rank. The construction uses "almost Diophantine quintuple"Recently, Drazic [466] significantly improved this result and showed that, assuming the Parity Conjecture for twists of certain elliptic curves, the density of q ∈ Q such that there exist infinitely many rational D(q)quintuples is at least 0.995.
On the other hand, in [78] a rational quintuple
9, 17/8, 27/10, 27/40, 493/40
with the same property was found, and the question arised whether there exists such quintuple consisting of positive rationals. The answer was given in [108] where it was proved that there exist infinitely many such quintuples. The "simplest" among them is12253738824071768160902809331272805381 / 13356284738726537361337339615814680856,
40228062558134597846809398333 / 2027377666049252712575626072,
90410203607675775632231738735 / 2640165528414654368852526998,
1459249660815833141719920182753327588589 / 13356284738726537361337339615814680856,
16463478877068761615 / 200378051669604563.
29/24, 71/54, 79/675, 1637/216, 2911/200.
It is clear that a set
Definition 5.1: A set of m nonzero rationals
{a_{1},
a_{2}, ... ,
a_{m}}
is called a strong Diophantine mtuple if

An example of a strong Diophantine triple is
{1976/5607, 3780/1691, 14596/1197}
In 2008, Dujella & Petricevic [162] proved that there exist infinitely many strong Diophantine triples. Also, there exist inifitely many such triples consisting of positive rationals.They found an example of an "almost" strong Diophantine quadruple:
{140/51, 2223/30464, 278817/33856. 3182740/17661},
which satisfies almost all conditions for a strong Diophantine quadruple. The only missing condition is that278817/33856 · 3182740/17661 + 1
is not a perfect square. However, no example of a strong Diophantine quadruple is known.In 2018, Dujella, Gusic, Petricevic & Tadic [381] considered strong Eulerian triples and proved that there exist infinitely many rationals a, b, c with the property that a^{2}  1, b^{2}  1, c^{2}  1, ab  1, ac  1 and bc  1 are all perfect squares.
In 2020, Dujella, Paganin & Sadek [429] proved that for infinitely many squarefree integers q (e.g. for q = 19, 17, 11, 7, 6, 5, 2, 6, 10, 13, 15, 17) there exist infinitely many rationals a, b, c with the property that a^{2} + q, b^{2} + q, c^{2} + q, ab + q, ac + q and bc + q are all perfect squares.
1. Introduction
2. Diophantine quintuple conjecture
3. Sets with the property D(n)
4. Connections with Fibonacci numbers
6. Connections with elliptic curves
7. Various generalizations
8. References
Diophantine mtuples page  Andrej Dujella home page 