5. Rational Diophantine m-tuples


5.1. Regular Diophantine quintuples

Euler proved that arbitrary rational Diophantine pair can be extended to a rational Diophantine quintuple. Euler's construction was generalized to rational Diophantine triples by Arkin, Hoggatt & Strauss [14] and to Diophantine quadruples by Dujella [62].

Let {a, b, c, d} be a rational Diophantine quadruple, i.e.

ab + 1 = r12,   ac + 1 = r22,   ad + 1 = r32,

bc + 1 = r42,   bd + 1 = r52,   cd + 1 = r62.  

Assume that abcd ≠ 1 and define

e = ((a+b+c+d)(abcd+1) + 2abc + 2abd + 2acd+2bcd ± 2r1r2r3r4r5 r6) / (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 = (ar4r5r6 ± r1r2r3)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

e1 = 549120 / 10201,   e2 = -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+b-c-d)2 - 4(ab+1)(cd+1)) / (4d(ab+1)(ac+1)(bc+1)).

5.2. Rational Diophantine sextuples

Although it was known already to Euler that there exist infinitely many rational Diophantine quintuples, until recently it was not known whether there exist infinitely many rational Diophantine sextuples, and it is still not known whether there exist any rational Diophantine septuple (7-tuple). Gibbs [144, 80] found 45 examples of rational Diophantine sextuples (they can be seen here).

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(t2-1) / ((t2-6t+1)(t2+6t+1)),
b = (t-1)(t2+6t+1)2 / (6t(t+1)(t2-6t+1)),
c = (t+1)(t2-6t+1)2 / (6t(t-1)(t2+6t+1)),
d = d1 / d2, where
d1 = 6(t+1)(t-1)(t2+6t+1)(t2-6t+1)(8t6+27t5+24t4-54t3+24t2+27t+8)(8t6-27t5+24t4+54t3+24t2- 27t+8)(t8+22t6-174t4+22t2+1),
d2 = t(37t12-885t10+9735t8-13678t6+9735t4-885t2+37)2,
e = e1 / e2, where
e1 = -2t(4t6-111t4+18t2+25)(3t7+14t6-42t5+30t4+51t3+18t2-12t+2)(3t7-14t6-42t5- 30t4+51t3-18t2-12t-2)(t2+3t-2)(t2-3t-2)(2t2+3t-1)(2t2-3t-1)(t2+7)(7t2+1),
e2 = 3(t+1)(t2-6t+1)(t-1)(t2+6t+1)(16t14+141t12-1500t10+7586t8-2724t6+165t4+424t2-12)2,
f = f1 / f2, where
f1 = 2t(25t6+18t4-111t2+4)(2t7-12t6+18t5+51t4+30t3-42t2+14t+3)(2t7+12t6+18t5- 51t4+30t3+42t2+14t-3)(2t2+3t-1)(2t2-3t-1)(t2-3t-2)(t2+3t-2)(t2+7)(7t2+1),
f2 = 3(t+1)(t2-6t+1)(t-1)(t2+6t+1)(12t14-424t12-165t10+2724t8-7586t6+1500t4-141t2-16)2.

Note that for 5.83 < t < 6.86 all six numbers a,b,c,d,e,f are positive.

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 = (t2-2t-1)(t2+2t+3)(3t2-2t+1) / (4t(t2-1)(t2+2t-1)),
b = 4t(t2-1)(t2-2t-1) / ((t2+2t-1)3),
c = 4t(t2-1)(t2+2t-1) / ((t2-2t-1)3),
d = (t2+2t-1)(t2-2t+3)(3t2+2t+1) / (4t(t2-1)(t2-2t-1)),
e = (-t5+14t3-t) / (t6-7t4+7t2-1),
f = (3t6-13t4+13t2-3) / (4t(t4-6t2+1)).

Dujella, Kazalicki & Petricevic [406] proved that there are infinitely many rational Diophantine sextuples such that denominators of all the elements (in the lowest terms) in the sextuples are perfect squares. Their motivating example was

{75/82, -3325/642, -12288/1252, 123/102, 3498523/22602, 698523/22602}.

In 2024, Rathbun found two pairs of such sextuples sharing a common triple:

{3/22, -12/52, -127500/3892, -2093/502, 72512/2652, -271859133/279702},
{3/22, -12/52, -127500/3892, 2387/502, 435252/6852, -38178413/61302}

and

{3/22, 123/82, 963963/4722, -2048/1312, -333333/17842, 265444352/59812},
{3/22, 123/82, 963963/4722, 15872/612, -79373/7062, 3434399232/481012}.

In [409], Dujella, Kazalicki & Petricevic proved that there are infinitely many rational Diophantine sextuples which contain one regular Diophantine quintuple and two regular Diophantine quadruples.

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.

In 2018, Dujella, Kazalicki & Petricevic found many examples with mixed signs, e.g.

{-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;

and also three examples with positive elements:

{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.

They also found an example of rational Diophantine quadruple which can be extended to rational Diophantine sextuples on three different ways:

{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 of

1519/408, 4505/168, 1959335/7824984, 13303605/1077512, 73026883629/17054089928, 515358540182255/7116911275416.

5.3. Rational D(q)-m-tuples

Let q be a rational number. A set of m non-zero rationals {a1, a2, ... , am} is called a rational D(q)-m-tuple if ai aj + q is a square of a rational number for all 1 ≤ i < j ≤ m.

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 q = r2 (see Chapter 5.1), q = -r2 ([108], see Chapter 5.4) and q = -3r2 (using that fact that the elliptic curve

y2 = x3 + 42x2 + 432x + 1296

has positive rank, see [88]).

In 2012, Dujella & Fuchs [238] obtained much more general result by proving that for infinitely many square-free numbers q there are infinitely many rational D(q)-quintuples, by considering twists of the elliptic curve

y2 = x3 + 86x2 + 825x

with positive rank. The construction uses "almost Diophantine quintuple" {x, 9x + 8, 25x + 20, 4x + 2, 16x + 14}, i.e. quintuple of linear polynomials containing two polynomial D(10x + 9)-quadruples (see also [140]).

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.

5.4. A problem of Diophantus and Euler

Diophantus found three rationals 3/10, 21/5, 7/10 with the property that the product of any two of them increased by the sum of those two gives a perfect square. Euler found four rationals 5/2, 9/56, 65/224, 9/224 with the same property and asked if there is an integer solution of this problem. In 2005, Dujella & Fuchs [131] proved that there does not exist a set of four positive integers with the same property.

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 is 
  12253738824071768160902809331272805381 / 13356284738726537361337339615814680856, 
 
        40228062558134597846809398333 / 2027377666049252712575626072, 

        90410203607675775632231738735 / 2640165528414654368852526998, 
 
1459249660815833141719920182753327588589 / 13356284738726537361337339615814680856, 

     16463478877068761615 / 200378051669604563.
A smaller example has been found by Petricevic:

29/24, 71/54, 79/675, 1637/216, 2911/200.

It is clear that a set {x1, x2, ... , xm} of m rationals has the property that xixj + xi + xj is a perfect square for all 1 ≤ i < j ≤ m if and only if {x1+1, x2+1, ... , xm+1} is a rational D(-1)-m-tuple. Therefore, the above mentioned result from [108] shows that there exist infinitely many rational D(-1)-quintuples, and consequently there exist infinitely many rational D(-q2)-quintuples for every rational number q.

5.5. Strong Diophantine triples

Note that in the definition of (rational) Diophantine m-tuples we excluded i = j, i.e. the condition that ai2 + 1 is a perfect square. It is obvious that for integers such condition has no sense. But for rationals there is no obvious reason why the sets which satisfy these stronger conditions would not exist.

Definition 5.1: A set of m nonzero rationals {a1, a2, ... , am} is called a strong Diophantine m-tuple if ai · aj + 1 is a perfect square for all i, j = 1, 2, ..., m.

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 that

278817/33856 · 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 a2 - 1, b2 - 1, c2 - 1, ab - 1, ac - 1 and bc - 1 are all perfect squares.

In 2020, Dujella, Paganin & Sadek [429] proved that for infinitely many square-free 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 a2 + q, b2 + q, c2 + 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 m-tuples page Andrej Dujella home page