3. Sets with the property D(n)

3.1. Diophantine quadruples with property D(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 < j ≤ m. 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 a_i a_j ≡ 2 or 3 (mod 4). It implies that none of the a_i is divisible by 4. Therefore, we may assume that a₁ ≡ a₂ (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 n ∉ S = {-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 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.

{1, 9k² + 8k + 1, 9k² + 14k +6, 36k² + 44k + 13}

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 N^1/3 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)

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

M_n = sup {|S| : S has the property D(n)},

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 M_n 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

11.006 log|n|

By improving estimates for the number of very small elements, Becker and Ram Murty [398] proved that M_n < 2.6071 log|n| holds for sufficiently large |n|.

In 2005, Dujella and Luca [132] proved that M_p < 3 ^. 2¹⁶⁸ 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.

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