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 P_{n}^{(k)}-set of size m
is a set {a_{1},
a_{2}, ... , a_{m}}
of distinct positive integers such that
∏_{j∈J}a_{j} + n
is a k-th power of an integer, for each
J ⊆
{1, 2, ..., m} where |J| = k.
They proved that any P_{n}^{(k)}-set
is finite.
7.2. Polynomials
Let n be a polynomial with integer coefficients. Let
D = {a_{1}, a_{2}, ... ,
a_{m}} 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
a_{1}/p, a_{2}/p, ... ,
a_{m}/p and n/p^{2}
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
P_{n} =
sup {|S| : S is a polynomial D(n)-tuple}.
Theorem 2.2 implies that P_{1} = 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 P_{n}
≤ 22 for all polynomials
n of degree 0. These results also give a bound for
P_{n} in terms of the degree and the
maximum of the coefficients of n.
It would be interesting to find an upper bound for
P_{n} which depends only on degree of
n. This was done for linear polynomials by Dujella, Fuchs,
Tichy and Walsh [109,
140], they proved that
P_{n}
≤ 12 for all polynomials n of degree 1,
and for quadratic polynomials by Jurasic [222],
she proved that P_{n}
≤ 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 {a_{1}, a_{2}, ... ,
a_{m}} is a set of polynomials, not all of them
constant, with coefficients in
K, with the property that
a_{i}a_{j} + 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 = {a_{1},
a_{2}, ... , a_{m}}
has a congruence type[b_{1},
b_{2}, ... , b_{m}],
where b_{i}
∈ {0, 1, 2, 3},
if a_{i}
≡ b_{i} (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.