Diophantine M-tuple Research Articles

IMPROVED UPPER BOUNDS ON DIOPHANTINE TUPLES WITH THE PROPERTY $D(n)$

Abstract Let n be a nonzero integer. A set S of positive integers is a Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for each $a,b \in S$ with $a \neq b$ . It is of special interest to estimate the quantity $M_n$ , the maximum size of a Diophantine tuple with the property $D(n)$ . We show the contribution of intermediate elements is $O(\log \log |n|)$ , improving a result by Dujella [‘Bounds for the size of sets with the property $D(n)$ ’, Glas. Mat. Ser. III39(59)(2) (2004), 199–205]. As a consequence, we deduce that $M_n\leq (2+o(1))\log |n|$ , improving the best known upper bound on $M_n$ by Becker and Murty [‘Diophantine m-tuples with the property $D(n)$ ’, Glas. Mat. Ser. III54(74)(1) (2019), 65–75].

TESTING THE ELONGATION OF 3-TUPLES RELATING TO POLYNOMIALS AND POLYGONAL NUMBERS WITH PECULIAR PROCLAMATION

In this paper, special 2-tuples with elements stand well-known polynomials and polygonal numbers with different sides that exclusively meet the property that the summation or difference of two elements provides a square can be stretched to 3-tuples but not 4-tuples with the same characteristics are displayed. Also, the similar characteristics of all such 3-tuples are checked with numerical values by Python programs. KEYWORDS: Diophantine m-tuples, Congruence relations, polygonal numbers, polynomials

On the number of Diophantine m-tuples in finite fields

We use a new argument to improve the error term in the asymptotic formula for the number of Diophantine m-tuples in finite fields, which is due to A. Dujella and M. Kazalicki (2021) and N. Mani and S. Rubinstein-Salzedo (2021).

On the extensions of the Diophantine triples in Gaussian integers

A Diophantine m-tuple is a set of m distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine triple \(\{k-1, k+1, 16k^3-4k\}\) in Gaussian integers \({\mathbb {Z}}{[i]}\) to a Diophantine quadruple. Similar one-parameter family, \(\{k-1, k+1, 4k\}\), was studied in Franušić (Glasnik matematički 43(2):265–291, 2008), where it was shown that the extension to a Diophantine quadruple is unique (with an element \(16k^3-4k\)). The family of the triples of the same form \(\{k-1, k+1, 16k^3-4k\}\) was studied in rational integers in Bugeaud et al. (Glasg Math J 49:333–344, 2007). It appeared as a special case while solving the extensibility problem of Diophantine pair \(\{k-1, k+1\}\), in which it was not possible to use the same method as in the other cases. As authors (Bugeaud, Dujella and Mignotte) point out, the difficulty appears because the gap between \(k+1\) and \(16k^3-4k\) is not sufficiently large. We find the same difficulty here while trying to use Diophantine approximations. Then we partially solve this problem by using linear forms in logarithms.

Diophantine triples and K3 surfaces

A Diophantine m-tuple with elements in the field K is a set of m non-zero (distinct) elements of K with the property that the product of any two distinct elements is one less than a square in K. Let X:(x2−1)(y2−1)(z2−1)=k2, be an affine variety over K. Its K-rational points parametrize Diophantine triples over K such that the product of the elements of the triple that corresponds to the point (x,y,z,k)∈X(K) is equal to k. We denote by X‾ the projective closure of X and for a fixed k by Xk a variety defined by the same equation as X.In this paper, we try to understand what can the geometry of varieties Xk, X and X‾ tell us about the arithmetic of Diophantine triples.First, we prove that the variety X‾ is birational to P3 which leads us to a new rational parametrization of the set of Diophantine triples.Next, specializing to finite fields, we find a correspondence between a K3 surface Xk for a given k∈Fp× in the prime field Fp of odd characteristic and an abelian surface which is a product of two elliptic curves Ek×Ek where Ek:y2=x(k2(1+k2)3+2(1+k2)2x+x2). We derive an explicit formula for N(p,k), the number of Diophantine triples over Fp with the product of elements equal to k. Moreover, we show that the variety X‾ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on X‾ over an arbitrary finite field Fq. Using it we reprove the formula for the number of Diophantine triples over Fq from [DK21].Curiously, from the interplay of the two (K3 and rational) fibrations of X‾, we derive the formula for the second moment of the elliptic surface Ek (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating S4(Γ0(8)).Finally, in the Appendix, Luka Lasić defines circular Diophantine m-tuples, and describes the parametrization of these sets. For m=3 this method provides an elegant parametrization of Diophantine triples.

Rational Diophantine sextuples containing two regular quadruples and one regular quintuple

A set of m distinct nonzero rationals {a1,a2,…,am} such that aiaj+1 is a perfect square for all 1 ≤ i < j ≤ m, is called a rational Diophantine m-tuple. It is proved recently that there are infinitely many rational Diophantine sextuples. In this paper, we construct infinite families of rational Diophantine sextuples with special structure, namely the sextuples containing quadruples and quintuples of certain type.

Diophantine m-tuples in finite fields and modular forms

For a prime p, a Diophantine m-tuple in $$\mathbb {F}_p$$ is a set of m nonzero elements of $$\mathbb {F}_p$$ with the property that the product of any two of its distinct elements is one less than a square. In this paper, we present formulas for the number $$N^{(m)}(p)$$ of Diophantine m-tuples in $$\mathbb {F}_p$$ for $$m=2,3$$ and 4. Fourier coefficients of certain modular forms appear in the formula for the number of Diophantine quadruples. We prove that asymptotically $$N^{(m)}(p)=\displaystyle \frac{1}{2^{m \atopwithdelims ()2 }}\frac{p^m}{m!} + o(p^m)$$ , and also show that if $$p>2^{2m-2}m^2$$ , then there is at least one Diophantine m-tuple in $$\mathbb {F}_p$$ .

The Paley graph conjecture and Diophantine m-tuples

A Diophantine m-tuple with property D(n), where n is a non-zero integer, is a set of m positive integers {a1,...,am} such that aiaj+n is a perfect square for all 1⩽i<j⩽m. It is known that Mn=sup⁡{|S|:S is a D(n) m-tuple} exists and is O(log⁡|n|). In this paper, we show that the Paley graph conjecture implies that the upper bound can be improved to ≪(log⁡|n|)ϵ, for any ϵ>0.

Rational Diophantine sextuples with square denominators

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and in 2016 Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many of them. In this paper, we prove that there exist infinitely many rational Diophantine sextuples such that the denominators of all the elements in the sextuples are perfect squares.

Some family of Diophantine pairs with Fibonacci numbers

A set {a1, a2, …, am} of positive integers is called a Diophantine m-tuple if aiaj + 1 is a perfect square for all 1 ≤ i < j ≤ m. In this paper, we prove that if {F2k, F2k+4, c} is a Diophantine triple then there is a unique positive integer d such that c < d and {F2k, F2k+4, c, d} is a Diophantine quadruple.

Triples which are D(n)-sets for several n's

For a nonzero integer n, a set of distinct nonzero integers {a1,a2,…,am} such that aiaj+n is a perfect square for all 1≤i<j≤m, is called a Diophantine m-tuple with the property D(n) or simply D(n)-set. D(1)-sets are known simply as Diophantine m-tuples. Such sets were first studied by Diophantus of Alexandria, and since then by many authors. It is natural to ask if there exists a Diophantine m-tuple (i.e. D(1)-set) which is also a D(n)-set for some n≠1. This question was raised by Kihel and Kihel in 2001. They conjectured that there are no Diophantine triples which are also D(n)-sets for some n≠1. However, the conjecture does not hold, since, for example, {8,21,55} is a D(1) and D(4321)-triple, while {1,8,120} is a D(1) and D(721)-triple. We present several infinite families of Diophantine triples {a,b,c} which are also D(n)-sets for two distinct n's with n≠1, as well as some Diophantine triples which are also D(n)-sets for three distinct n's with n≠1. We further consider some related questions.

What is a Diophantine $m$-tuple?

The problem of the construction of Diophantine m-tuples, i.e. sets with the property that product of any two of its distinct elements is one less then a square, has a very long history. There are some new results in this area, but many open problems and unproved conjectures still remains. In this survey we explain the main problems and results concerning Diophantine m-tuples.

There Are Infinitely Many Rational Diophantine Sextuples

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. In this paper, we prove that there exist infinitely many rational Diophantine sextuples.

Bounds for Diophantine quintuples II

A set of positive integers {; ; a1, a2, ..., a_m}; ; with the property that a_i*a_j+1 is a perfect square for all distinct indices i and j between 1 and m is called Diophantine m-tuple. In this paper, we show that if {; ; a, b, c, d, e}; ; is a Diophantine quintuple with a 3ag ; moreover, if c > a + b + 2\sqrt{; ; ab + 1}; ; then b > max{; ; 24 ag, 2a^1.5g^2}; ; . Similar results are given assuming that either ab is odd or c = a + b + 2\sqrt{; ; ab + 1}; ; .

Gaussian -Diophantine quadruples with property D (1)

A set of m Gaussian integers is called a complex Diophantine m-tuple with the property D (z) if the product of its any two distinct elements increased by z is a square of a Gaussian integer. In this paper, we present Gaussian-Diophantine quadruples with property D(1).Few examples of complex Diophantine quadruples with the property D (1) are presented.

Diophantine [formula omitted]-tuples with elements in arithmetic progressions

In this paper, we consider the problem of existence of Diophantine m-tuples which are (not necessarily consecutive) elements of an arithmetic progression. We show that for n≥3 there does not exist a Diophantine quintuple {a,b,c,d,e} such that a≡b≡c≡d≡e(modn). On the other hand, for any positive integer n there exist infinitely many Diophantine triples {a,b,c} such that a≡b≡c≡0(modn).

Diophantine m-tuples for quadratic polynomials

Abstract The poster presents one polynomial variant of the problem of Diophantus, described by A. Jurasic [Diophantine m-tuples for quadratic polynomials, Glas. Mat. Ser. III 46 (2011), 283–309], and ilustrates that results with some examples from the paper of A. Dujella and A. Jurasic [Some Diophantine triples and quadruples for quadratic polynomials, J. Comb. Number Theory 3(2) (2011), 123–141]. We proved that there does not exist a set with more than 98 nonzero polynomials in Z [ X ] , such that the product of any two of them plus a quadratic polynomial n is a square of a polynomial from Z [ X ] (we exclude the possibility that all elements of such set are constant multiples of a linear polynomial p ∈ Z [ X ] such that p 2 | n ). Specially, we prove that if such a set contains only polynomials of odd degree, then it has at most 18 elements.

The number of Diophantine quintuples II

A set {a1; : : : ; am} of m distinct positive integers is called a Diophantine m-tuple if aiaj + 1 is a perfect square for all i, j with 1 � i < jm. It is known that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. In this paper, we first show that for a fixed Diophantine triple {a; b; c} with a < b < c, the number of Diophantine quintuples {a; b; c; d; e} with c < d < e is at most four. Using this result, we further show that the number of Diophantine quintuples is less than 10276, which improves the bound 101930 due to Dujella.

On the size of sets whose elements have perfect power $n$-shifted products

Abstract. We show that the size of sets Ahaving the property that with somenon-zero integer n, a 1 a 2 + n is a perfect power for any distinct a 1 ;a 2 2A, cannot bebounded by an absolute constant. We give a much more precise statement as well,showing that such a set Acan be relatively large. We further prove that under the abc-conjecture a bound for the size of Adepending on n can already be given. Extending aresult of Bugeaud and Dujella, we also derive an explicit upper bound for the size of Awhen the shifted products a 1 a 2 + n are k-th powers with some xed k 2. The latterresult plays an important role in some of our proofs, too. 1. IntroductionA set A= fa 1 ;:::;a m gof positive integers is called a Diophantine m-tuple,if for any 1 i < j mwe have a i a j + 1 = x 2ij for an integer x ij . The Mathematics Subject Classi cation: 11B75, 11D99.Key words and phrases: shifted product, perfect power, abc-conjecture, Diophantine m-tuple.The rst three authors are supported by the Hungarian-Croatian bilateral project Numbertheory and cryptography. The rst and third authors are supported in part by the OTKA grantsK67580 and K75566, and by the TAMOP 4.2.1./B-09/1/KONV-2010-0007 project. The projectis implemented through the New Hungary Development Plan, co nanced by the European SocialFund and the European Regional Development Fund. The second author is supported by theMinistry of Science, Education and Sports, Republic of Croatia, grant 037-0372781-2821. Thefourth author was supported in part by Grants SEP-CONACyT 79685 and PAPIIT 100508.

Diophantine m-tuples for quadratic polynomials

Abstract The poster presents one polynomial variant of the problem of Diophantus, described by A. Jurasic [Diophantine m-tuples for quadratic polynomials, Glas. Mat. Ser. III 46 (2011), 283–309], and ilustrates that results with some examples from the paper of A. Dujella and A. Jurasic [Some Diophantine triples and quadruples for quadratic polynomials, J. Comb. Number Theory 3(2) (2011), 123–141]. We proved that there does not exist a set with more than 98 nonzero polynomials in Z [ X ] , such that the product of any two of them plus a quadratic polynomial n is a square of a polynomial from Z [ X ] (we exclude the possibility that all elements of such set are constant multiples of a linear polynomial p ∈ Z [ X ] such that p 2 | n ). Specially, we prove that if such a set contains only polynomials of odd degree, then it has at most 18 elements.