Definition 3.1: Let n be an integer. A set of
m positive integers
|
Several authors considered the problem of the existence of Diophantine quadruples with the property D(n). This problem is almost completely solved. In 1985, Brown [21], Gupta & Singh [22] and Mohanty & Ramasamy [24] proved independently the following result, which gives the first part of the answer.
Theorem 3.1: If n is an integer of the form n = 4k + 2, then there does not exist a Diophantine quadruple with the property D(n). |
The proof of Theorem 3.1 is very simple. Indeed, assume that
In 1993, Dujella [44] gave the second part of the answer.
Theorem 3.2: If an integer n does not have the form
4k + 2 and n
∉
S = |
The conjecture is that for n ∈ S there does not exist a Diophantine quadruple with the property D(n).
For n = -1, there are results which show that some particular Diophantine triples cannot be extended to quadruples [20, 21, 28, 71, 75, 92, 116, 131, 134, 139, 150, 153, 156, 176, 205, 223, 229, 234]. Dujella & Fuchs [131] proved that there does not exist a Diophantine quintuple with the property D(-1), and Dujella, Filipin & Fuchs [150] proved that there are only finitely many such quadruples.
Recent work of Bonciocat, Cipu, and Mignotte [449] establish the non-existence of D(-1)-quadruples. The proof is based on several new ideas and combines in an innovative way techniques proved successful in dealing with D(n)-sets with less usual tools, developed for the study of different problems. Note that this result entails the non-existence of D(-4)-quadruples, because in [44] it was shown that all elements of a D(-4)-quadruple are even.
Theorem 3.2 was proved by considering the following six cases:
n = 4k + 3, n = 8k + 1, n = 8k + 5, n = 8k, n = 16k + 4, n = 16k + 12.
In each of these cases, it is possible to find a set with the property D(n) consisted of the four polynomials in k with integer coefficients. For example, the set{1, 9k2 + 8k + 1, 9k2 + 14k +6, 36k2 + 44k + 13}
has the property D(4k + 3). The elements from the set S are exceptions because we can get the sets with nonpositive or equal elements for some values of k.
Formulas of the similar type were systematically derived in
[56].
Using these formulas, in [69] and
[70], some improvements
of Theorem 3.2
were obtained. It was proved that if
|n| is sufficiently large and
Let U denote the set of all integers n, not of the form 4k + 2, such that there exist at most two distinct Diophantine quadruples with the property D(n). An open question is whether the set U is finite or not.
There are infinitely many D(1)-quadruples
(e.g. {k - 1, k + 1, 4k, 16k3 - 4k} for k ≥ 2).
More precisely, it was proved in [195] that the number of D(1)-quadruples with elements
≤ N is
If n is a perfect square, say n = k2, then by multiplying elements of a D(1)-quadruple by k we obtain a D(k2)-quadruple, and thus we conclude that there exist infinitely many D(k2)-quadruples. The following conjecture was proposed in [180].
Conjecture 3.1: If a nonzero integer n is not a perfect square, then there exist only finitely many D(n)-quadruples. |
The conjecture is known to be true for n ≡ 2 mod 4, n = -1 and n = -4.
Motivated with this conjecture, in [411]
it was considered the question, for given integer n which is not a perfect square,
what can be said about the largest element in a D(n)-quadruple.
Let {a, b, c, d} be a D(n)-quadruple such that
|d| is the maximal absolute value. It is shown that the set of possible
Let n be a nonzero integer. We may ask how large a set with the property D(n) can be. Let define
Mn = sup {|S| : S has the property D(n)},
where |S| denotes the number of elements in the set S.
By the results of Chapter 2 we know that
Dujella [107, 123] proved that Mn is finite for all n. More precisely, it holds:
Theorem 3.3:
Mn
≤ 31 for
|n| ≤ 400, |
In the proof of Theorem 3.3, the numbers of "large" (greater than |n|3), "small" (between n2 and |n|3) and "very small" (less than n2) elements were estimated separately. Using a theorem of Bennett on simultaneous approximations of algebraic numbers and a gap principle, it was proved that the number of large elements is less than 22 for all nonzero integers n. For the estimate of the number of small elements, a weak variant of the gap principle was used to prove that this number is less than
0.6114 log|n| + 2.158
for all nonzero integers n (and less than11.006 log|n|
for |n| > 400. It is easy to check that there are at most 5 very small elements forBy improving estimates for the number of very small elements, Becker and Ram Murty [398] proved that Mn < 2.6071 log|n| holds for sufficiently large |n|.
In 2005, Dujella and Luca [132] proved that Mp < 3 . 2168 holds for all primes p. Furtermore, they showed that Mn is bounded in terms of the number of prime factors of n for squarefree values of n. They also showed that for almost all n (in the sense of natural density), the estimate Mn < log log |n| holds.
In 2001, A. Kihel & O. Kihel [102] possed the following question:
Is there any Diophantine triple (i.e. D(1)-triple) which is also a D(n)-triple for some n ≠ 1?
Zhang & Grossman [335] found triple {1, 8, 120} which is a D(1) and D(721)-triple.
Adzaga, Dujella, Kreso & Tadic [360] proved that there exist
infinitely many D(1)-triples which are also D(n)-triples for two distinct
n's with n ≠ 1. More precisely, they showed that
if
They also found several examples of Diophantine triples which are D(n)-triples for three distinct n's with n ≠ 1. E.g. {4, 12, 420} is a D(1), D(436), D(3796) and D(40756)-triple. An open question is whether there are there infinitely many such triples.
Dujella & Petricevic [403] showed that there are infinitely many nonequivalent (i.e. nonproportional) sets of four distinct nonzero integers {a, b, c, d} with the property that there exist two distinct nonzero integers n1 and n2 such that {a, b, c, d} is a D(n1)-quadruple and a D(n2)-quadruple. E.g. {-1, 7, 119, 64} is a D(128) and D(848)-quadruple, while {15, 380, 5735, 634880} is a D(361536) and D(7123200)-quadruple.
In [430], they showed that there are infinetely many such quadruples consisting of perfect squares (so they are also D(0)-quadruples). E.g. {2916, 132496, 10000, 28224} is a D(67076100), D(1625702400) and D(0)-quadruple, while {2133516100, 14428814400, 16048835856, 88439622544} is a D(361870328733788160000), D(120743569936436464836) and D(0)-quadruple.
Furthermore, in [452], Dujella, Kazalicki & Petricevic showed that there are infinitely many quintuples consisting of perfect squares which are D(n)-quintuples for certain non-zero integer n. E.g. {50625, 1982464, 670761, 81796, 6492304} is a D(230861030400) and D(0)-quintuple, while {464265042831500625, 38320173497073664, 1348817972910840441, 68867272163656516, 4875819601620659344} is a D(77290653706850480516933191188710400) and D(0)-quintuple.
1. Introduction
2. Diophantine quintuple conjecture
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 |