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] has given the second part of the answer.
Theorem 3.2: If an integer n does not have the form
4k + 2 and n
∉
S = 
For n ∈ S, the question of the existence of Diophantine quadruples with the property D(n) is still open. The conjecture is that for these values of n there does not exist a Diophantine quadruple.
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.
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, 9k^{2} + 8k + 1, 9k^{2} + 14k +6, 36k^{2} + 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, 16k^{3}  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 = k^{2}, then by multiplying elements of a D(1)quadruple by k we obtain a D(k^{2})quadruple, and thus we conclude that there exist infinitely many D(k^{2})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
M_{n} = 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 M_{n} is finite for all n. More precisely, it holds:
Theorem 3.3:
M_{n}
≤ 31 for
n ≤ 400, 
In the proof of Theorem 3.3, the numbers of "large" (greater than n^{3}), "small" (between n^{2} and n^{3}) and "very small" (less than n^{2}) 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 logn + 2.158
for all nonzero integers n (and less than11.006 logn
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 M_{n} < 2.6071 logn holds for sufficiently large n.
In 2005, Dujella and Luca [132] proved that M_{p} < 3 ^{.} 2^{168} holds for all primes p. Furtermore, they showed that M_{n} 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 M_{n} < 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 n_{1} and n_{2} such that {a, b, c, d} is a D(n_{1})quadruple and a D(n_{2})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, there are infinitely many quintuples consisting of perfect squares which are D(n)quintuples for certain nonzero 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 mtuples
6. Connections with elliptic curves
7. Various generalizations
8. References
Diophantine mtuples page  Andrej Dujella home page 