## 2. Diophantine quintuple conjecture

In 1979 Arkin, Hoggatt and Strauss [14] proved that every Diophantine triple can be extended to a Diophantine quadruple. More precisely, let {a, b, c} be a Diophantine triple and

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

where r,s,t are positive integers. Define

d+ = a + b + c + 2abc + 2rst.

Then {a, b, c, d+} is a Diophantine quadruple. Indeed,

ad+ + 1 = (at + rs)2,     bd+ + 1 = (bs + rt)2,     cd+ + 1 = (cr + st)2.

Now we can give a stronger version of the Diophantine quintuple conjecture.

 Conjecture 2.1: If {a, b, c, d} is a Diophantine quadruple and d > max {a, b, c}, then d = d+.

It is clear that Conjecture 2.1 implies that there does not exist a Diophantine quintuple.

As we mentioned in Chapter 1, Baker & Davenport [3] verified Conjecture 2.1 for the Diophantine triple {1, 3, 8}. Veluppillai [18] verified the conjecture for the triple {2, 4, 12} and Kedlaya [75] for the triples {1, 3, 120}, {1, 8, 120}, {1, 8, 15}, {1, 15, 35}, {1, 24, 35} and {2, 12, 24}. In [64] and [77], the conjecture was verified for all triples of the form {k - 1, k + 1, 4k} and {F2k, F2k+2, F2k+4}, respectively. Furthemore, in [73], Dujella & Pethő proved that the pair {1, 3} cannot be extended to a Diophantine quintuple. In 2008, Fujita [168] (see also [148]) proved that for k ≥ 2, the Diophantine pair {k - 1, k + 1} cannot be extended to a Diophantine quintuple.

A Diophantine quadruple D = {a, b, c, d}, where a < b < c < d, is called regular if d = d+. Equivalently, D is regular iff

(a + b - c - d)2 = 4(ab + 1)(cd + 1)

(see [80]). This equation is a quadratic equation in d. One root of this equation is d+, and the other root is

d_ = a + b + c + 2abc - 2rst.

It is easy to check that all "small" Diophantine quadruples are regular. E.g. there are exactly 207 quadruples with max {a, b, c, d} < 106 and all of them are regular.

Since the number of integer points on an elliptic curve

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

is finite, it follows that there does not exist an infinite set of positive integers with the property of Diophantus and Fermat. However, bounds for the size and for the number of solutions depend on a,b,c and, accordingly, they do not immediately yield an absolute bound for the size of such set. The first absolute bound (m ≤ 8) for the size of Diophantine m-tuples was given in 2001 by Dujella (see [96]). In 2004, this result was significantly improved in [122]. The main results of [122] are the following two theorems.

 Theorem 2.1: There does not exist a Diophantine sextuple.

 Theorem 2.2: There are only finitely many Diophantine quintuples.

Moreover, the result from Theorem 2.2 is effective. Namely, it was proved in [122] that all Diophantine quintuples Q satisfy max Q < 1010. This implies that there are at most 101930 Diophantine quintuples (see [160]). This bound was significantly improved by Fujita in [173] by showing that there exist at most 10276 Diophantine quintuples. Furthermore, Filipin & Fujita [235] obtained the bound 1096, Elsholtz, Filipin & Fujita [271] the bound 6.8 · 1032, Cipu [312] the bound 1031, Trudgian [314] the bound 2.3 · 1029, and Cipu & Trudgian [325] the bound 1.18 · 1027

Theorems 2.1 and 2.2 improve results from [96] where it is proved that there does not exist a Diophantine 9-tuple and that there are only finitely many Diophantine 8-tuples.

As in [96], the main idea was to prove Conjecture 2.1 for a wide class of Diophantine triples, namely, for triples satisfying some gap conditions. However, in [122] these gap conditions are much weaker then in [96]. Thus, the class of Diophantine triples for which Conjecture 2.1 can be proved is now so wide that in an arbitrary Diophantine quadruple (with sufficiently large elements), we may find a sub-triple belonging to that class. And this is just what is needed in order to prove Theorem 2.2.

In the proof of Conjecture 2.1 for a triple {a, b, c}, the problem is first transformed into solving systems of simultaneous Pellian equations. This reduces to finding intersection of binary recursive sequences. Next step is the determination of initial terms of these sequences, under assumption that they have nonempty intersection which induces a solution of the original problem. This part is considerable improvement of the corresponding part of [96]. This improvement is due to new "gap principles".

Applying some congruence relations modulo c2, lower bounds for solutions are obtained. In obtaining these bounds, it is necessary to assume that our triple satisfies some gap conditions, e.g. b > 4a and c > b2.5. Let us note that these conditions are much weaker then conditions used in [96], and this is due to more precise determination of the initial terms. The comparison of these lower bounds with upper bounds obtained from the Baker's theory on linear forms in logarithms of algebraic numbers (the theorem of Baker and Wüstholz, or more recent theorem of Matveev) yields Theorem 2.2, and the comparison with upper bounds obtained from a theorem of Bennett on simultaneous approximations of algebraic numbers yields Theorem 2.1.

Let us note that in order to apply Bennett's theorem, it is necessary to assume that there is a large gap between b and c (something like c > b5). This is the reason why this approach doesn't give explicit information about the number of Diophantine quintuples. But, when these gap conditions are satisfied, upper bounds for solutions obtained by Bennett's theorem (i.e. by hypergeometric method) are much better than those obtained using linear forms in logarithms, and it makes possible to do all necessary computations and to prove that there is no Diophantine sextuple.

We also mention a result by Fujita [172], who proved that any Diophantine quintuple contains a regular Diophantine quadruple, i.e. if {a, b, c, d, e} is a Diophantine quintuple and a < b < c < d < e, then d = d+.

From the results of [271] and [312] (see also [325]), it follows that any quintuple {a, b, c, d, e} with a < b < c < d < e must be of one of the following types:

• 4a < b   and   4ab + a + b < c < b3/2,
• 4a < b   and   c = a + b + 2√ab+1,
• 4a < b   and   c > b3/2,
• b < 4a   and   c = a + b + 2√ab+1.
Finally, in the paper announced in 2016 and published in 2019, He, Togbé and Ziegler [350] gave the proof of the Diophantine quintuple conjecture (Conjecture 1.1).

 Theorem 2.3: There does not exist a Diophantine quintuple.

The three new key arguments that lead to the proof are:

• the definition of an operator on Diophantine triples and their classification;
• the use of sharp lower bounds for linear forms in three logarithms obtained by applying a result due to Mignotte;
• the use of new congruences in the case of Euler quadruples {a, b, a+b+2r, 4r(r+a)(r+b)}.

Bliznac Trebjesanin and Filipin [361] extended Theorem 2.3 and showed that there does not exist a D(4)-quintuple.

Conjecture 2.1 still remains open. In that direction, Fujita and Miyazaki [328] proved that any fixed Diophantine triple can be extended to a Diophantine quadruple in at most 11 ways by joining a fourth element exceeding the maximal element in the triple, while Cipu, Fujita and Miyazaki [370] improved that result by replacing 11 with 8.

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