3. Sets with the property D(n)

3.1. Diophantine quadruples with property D(n)

There are several natural generalizations of the original problem of Diophantus and Fermat. The first of them is the replacement of number 1, in the definition of Diophantine m-tuples, by an arbitrary integer n.

Definition 3.1: Let n be an integer. A set of m positive integers {a1, a2, ... , am} is said to have the property D(n) if ai · aj + n is a perfect square for all 1 ≤ i < jm. Such a set is called a Diophantine m-tuple with the property D(n) (or D(n)-m-tuple, or Pn-set of size m).

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 {a1, a2, a3, a4} has the property D(n). Since the square of an integer is   ≡ 0 or 1 (mod 4), we have that ai aj ≡ 2 or 3 (mod 4). It implies that none of the ai is divisible by 4. Therefore, we may assume that a1a2 (mod 4). But now we have that a1 a2 ≡ 0 or 1 (mod 4), a contradiction.

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 nS = {-4, -3, -1, 3, 5, 8, 12, 20}, then there exist at least one Diophantine quadruple with the property D(n).

The conjecture is that for nS 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 n ≡ 1 (mod 8), or n ≡ 4 (mod 32), or n ≡ 0 (mod 16), then there exist at least six, and if n ≡ 8 (mod 16), or n ≡ 13, 21 (mod 24), or n ≡ 3, 7 (mod 12), then there exist at least four distinct Diophantine quadruples with the property D(n).

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 C N1/3 log N, where C ≈ 0.338285.

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 δ = (log|d|)/log n is dense in the interval [2/5, 3].

3.2. Diophantine quintuples with property D(n)

One may ask what is the least positive integer n1, and what is the greatest negative integer n2, for which there exist a Diophantine quintuple with the property D(ni), i = 1,2. It is known that n1 ≤ 256 and n2 ≥ -255, since the sets {1, 33, 105, 320, 18240} and {5, 21, 64, 285, 6720} have the property D(256), and the set {8, 32, 77, 203, 528} has the property D(-255) ([62, [68]).

3.3. Estimates for the size of Diophantine m-tuples

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 M1 = 4. Bliznac Trebjesanic and Filipin [361] proved analogous results for n = 4, i.e. M4 = 4.

Dujella [107, 123] proved that Mn is finite for all n. More precisely, it holds:

Theorem 3.3:

      Mn ≤ 31     for   |n| ≤ 400,
Mn < 15.476 log|n|     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 than 0.6071 log|n| + 2.152 for |n| > 400). Finally, in the estimate of the number of very small elements, a large sieve method due to Gallagher and a result of Vinogradov on double sums of Legendre's symbols were used to prove that this number is less than

11.006 log|n|

for |n| > 400. It is easy to check that there are at most 5 very small elements for n ≤ 400. Putting all these estimates together, we obtain Theorem 3.3.

By 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.

3.4. Triples and quadruples which are D(n)-sets for several n's

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 {2, a, b, c} is a regular Diophantine quadruple, then the Diophantine triple {a, b, c} is a D(n)-triple for two distinct n's with n ≠ 1.

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