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 = r^{2}, ac + 1 = s^{2}, bc + 1 = t^{2},
where r,s,t are positive integers. Defined_{+} = 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
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 isd_ = a + b + c + 2abc - 2rst.
It is easy to check that all "small" Diophantine quadruples are regular. E.g. there are exactly 207 quadruples withSince the number of integer points on an elliptic curve
y^{2} = (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 < 10^{10}.
This implies that there are at most 10^{1930} Diophantine quintuples
(see [160]). This bound was significantly improved by Fujita
in [173] by showing that there exist at most
10^{276} Diophantine quintuples. Furthermore,
Filipin and Fujita [235] obtained the bound 10^{96},
Elsholtz, Filipin and Fujita [271] the bound
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 c^{2},
lower bounds for solutions are obtained.
In obtaining these bounds, it is necessary to assume
that our triple satisfies
some gap conditions, e.g.
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
We also mention a result by Fujita [172],
who proved that any Diophantine quintuple contains a regular
Diophantine quadruple, i.e. if
From the results of [271] and [312]
(see also [325]), it follows
that any 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
Diophantine m-tuples page | Andrej Dujella home page |