## 7. Various generalizations

### 7.1. Higher powers

In [113], Bugeaud & Dujella considered the problem of the existence of sets of positive integers such that the product of any two of them increased by 1 is a k-th power, for an integer k ≥ 3. Such sets are called k-th power Diophantine tuples. Examples of such triples for k = 3 and k = 4 are given, respectively, by {2, 171, 25326} and {1352, 8539880, 9768370}. In [113], absolute upper bounds for the size of such sets were given.

 Theorem 7.1: Let k ≥ 3 be an integer and let C(k) = sup {|S| : S is a k-th power Diophantine tuple}. Then C(3) ≤ 7, C(4) ≤ 5, C(k) ≤ 4   for 5 ≤ k ≤ 176, and C(k) ≤ 3 for k ≥ 177.

A slightly more general problem has been considered by Gyarmati [99]. Let N and k ≥ 3 be positive integers. Let A and B be subsets of {1, 2, ... ,N} such that ab + 1 is a perfect k-th power whenever aA and bB. What can be said about the cardinalities of the sets A and B? Gyarmati proved that min {|A|, |B|} ≤ 1+(log log N)/log(k-1). In [113], Bugeaud & Dujella showed that min {|A|, |B|} ≤ 2 for k ≥ 177.

In [111], [121], [130], [138], [155] and [170] estimates for the size of a set D ⊆ {1, 2, ... , N} with the property that ab + 1 is a perfect power for all a, bD, ab, are given. The best known bound is due to Stewart [170]: |D| ≪ (log N)2/3 (log log N)1/3. Luca [138] showed that the abc-conjecture implies that |D| is bounded by an absolute constant.

In [101], A. Kihel & O. Kihel consired a different generalization of the problem of Diophantus and Fermat to higher powers. A Pn(k)-set of size m is a set {a1, a2, ... , am} of distinct positive integers such that jJ aj + n is a k-th power of an integer, for each J ⊆ {1, 2, ..., m} where |J| = k. They proved that any Pn(k)-set is finite.

### 7.2. Polynomials

Let n be a polynomial with integer coefficients. Let D = {a1, a2, ... , am} be a set of m nonzero polynomials with integer coefficients satisfying the condition that there does not exist a polynomial pZ[X] such that a1/p, a2/p, ... , am/p and n/p2 are integers. The set D is called a polynomial D(n)-m-tuple if the product of any two of its distinct elements increased by n is a square of a polynomial with integer coefficients.

A natural question is how large such sets can be. Let us define

Pn = sup {|S| : S is a polynomial D(n)-tuple}.

Theorem 2.2 implies that P1 = 4. Moreover, all polynomial D(1)-quadruples are regular, i.e. Conjecture 2.1 is valid for polynomials with integer coefficients (see [125]). In 2019, Filipin & Jurasic [364] proved that the same result is valid for polynomials with real coefficients. On the other hand, Dujella & Jurasic [196] showed that there are irregular D(1)-quadruples in polynomials with complex coefficients. Indeed, the D(1)-quadruple

{√(-3)/2, -2√(-3)/3 (x2 - 1), (-3 + √(-3))/3 x2 + 2√(-3)/3, (3 + √(-3))/3 x2 + 2√(-3)/3}

is not regular.

In [115], Dujella & Fuchs proved that P-1 = 3.

From the results of [107] (see Chapter 3.3) it follows that Pn ≤ 22 for all polynomials n of degree 0. These results also give a bound for Pn in terms of the degree and the maximum of the coefficients of n. It would be interesting to find an upper bound for Pn which depends only on degree of n. This was done for linear polynomials by Dujella, Fuchs, Tichy and Walsh [109, 140], they proved that Pn ≤ 12 for all polynomials n of degree 1, and for quadratic polynomials by Jurasic [222], she proved that Pn ≤ 98 for all polynomials n of degree 2.

Let us mention that a variant of the problem of Diophantus and Fermat for polynomials was first considered by Jones [5, 12]. He treated the classical case n = 1. Various polynomial Diophantine quadruples were systematically derived by Dujella [44, 56] and Ramasamy [51]. Here are some examples:

{4x, 25x + 1, 49x + 3, 144x + 8}   for n = 16x +1;
{4, 9x2 - 5x, 9x2 + 7x + 2, 36x2 + 4x}   for n = 8x + 1;
{2x + 3, 3x2 + 4x + 2, 9x2 + 10x + 3, 24x2 + 26x + 7}   for n = 9x4 + 6x3 - 19x2 - 20x - 5.

In [151], Dujella & Luca considered the higher power variant of the problem of Diophantus and Fermat for polynomials. Let K be an algebraically closed field of characteristic zero. They proved that for every k ≥ 3 there exist a constant P(k), depending only on k, such that if {a1, a2, ... , am} is a set of polynomials, not all of them constant, with coefficients in K, with the property that ai aj + 1 is a k-th power of an element of K[X] for 1 ≤ i < jm, then mP(k). More precisely, they proved that

m ≤ 5     if k = 3;
m ≤ 4     if k = 4;
m ≤ 3     for k ≥ 5;
m ≤ 2     for k even and k ≥ 8.

Furthermore, in [161], Dujella, Fuchs & Luca proved that m ≤ 10 if k = 2. They also obtained an absolute upper bound for the size of a set of polynomials with the property that the product of any two elements plus 1 is a perfect power.

### 7.3. Congruence types modulo 4

We say that a set of integers X = {a1, a2, ... , am} has a congruence type [b1, b2, ... , bm], where bi ∈ {0, 1, 2, 3}, if aibi (mod 4) for i = 1, 2, ..., m.

In [33], Mootha & Berzsenyi characterized congruence types modulo 4 of Diophantine triples having the property D(n) for some integer n. They proved that possible congruence types of Diophantine triples are

[0,0,0],   [0,0,1],   [0,0,2],   [0,0,3],   [0,1,1],   [0,1,3],   [0,2,2],

[0,3,3],   [1,1,1],   [1,1,2],   [1,2,3],   [2,2,2],   [2,3,3],   [3,3,3].

Starting with this result, in [72] congruence types modulo 4 of Diophantine quadruples and quintuples were characterized.

However, in order to get congruence types [1,1,1,1,1] and [3,3,3,3,3] in [72] it was necessary to allow the number 0 as an element of quintuples. Recently, Petricevic found examples with nonzero elements: {-273375, -361375, -504063, 833, 1377} is a D(831406275)-quintuple of the congruence type [1,1,1,1,1], {-9, 59, 6075, 47291, 555579} is a D(5117175)-quintuple of the congruence type [3,3,3,3,3].

### 7.4. Gaussian integers and integers in quadratic fields

Let z = a + bi be a Gaussian integer. A set of m Gaussian integers is called a complex Diophantine m-tuple with the property D(z) if the product of any two of its distinct elements increased by z is a square of a Gaussian integer. In [63], the problem of existence of complex Diophantine quadruples was considered.

It was proved that if b is odd or ab ≡ 2 (mod 4), then there does not exist a complex Diophantine quadruple with the property D(a + bi). It is interesting that this condition is equivalent to the condition that a + bi is not representable as a difference of the squares of two Gaussian integers. In that way, this result becomes an analogue of Theorem 3.1, since an integer n is of the form 4k + 2 iff n is not representable as a difference of the squares of two integers.

It was also proved that if a + bi is not of the above form and a + bi ∉ {2, -2, 1 + 2i, -1 - 2i, 4i, -4i}, then there exists at least one complex Diophantine quadruple with the property D(a + bi).

In [117], Abu Muriefah and Al- Rashed considered the analogous problem in the ring Z[√-2]. They proved that there exists a Diophantine quadruple with the property D(a + b√-2) if a and b satisfy some congruence conditions. Their result was improved in [199] and [260]. In [126], Franusic solved completely the analogous problem in the ring Z[√2]. She proved that there exist infinitely many Diophantine quadruples with the property D(z) if and only if z can be represented as a difference of two squares in Z[√2]. Analogous results for more general quadratic fields has been obtained by Franusic in [166, 167], and for certain cubic and quartic fields by Franusic [269] and Franusic & Jadrijevic [413].

Adzaga [394] proved that there is no Diophantine m-tuple with the property D(1) in the ring of integers of an imaginary quadratic field for m > 42.

1. Introduction
2. Diophantine quintuple conjecture
3. Sets with the property D(n)
4. Connections with Fibonacci numbers
5. Rational Diophantine m-tuples
6. Connections with elliptic curves
8. References