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(5)
≤ 5,
C(k)
≤ 4 for
6 ≤ 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
a ∈ A
and b ∈ B.
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, b
∈ D,a ≠ b, 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 [230], it was shown that for any positive integer m,
there is a positive integer n and a set of positive integers A such that
|A| ≥ m and ab + n
is a power of a positive integer for any a, b ∈ A, a ≠ b.
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
∏j∈Jaj + 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
p
∈
Z[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
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:
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
aiaj + 1 is
a k-th power of an element of
K[X]
for 1 ≤ i
< j ≤ m, then
m ≤ P(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 ai
≡ bi (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
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 possibility that n = 0.
Recently, Petricevic found examples with n ≠ 0:
{-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 a
≡ b ≡ 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].
However, in [490] Chakraborty, Gupta & Hoque showed that in
certain rings of the form Z[√4k+2] there are elements z
which are not difference of two squares but there exist a D(z)-quadruple
(explicit examples are given for z = 26 + 6√10 in Z[√10]
and z = 18 + 2√58 in Z[√58]).
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.