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

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, 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.

{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.

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 4 ≤ M₁ ≤ 5. Recently, Filipin [163, 164] proved analogous results for n = 4, i.e. 4 ≤ M₄ ≤ 5.

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|

Recently, Dujella and Luca [132] proved that M_p < 3 ^. 2¹⁶⁸ holds for all primes p.

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