Diophantine Quadruples Research Articles

Diophantine $$\mathcal{S} $$-quadruples with two primes which are twin

We show that there are only finitely many pairs of twin primes $$(p, p+2) $$ such that there exists an $$\mathcal{S} $$ -Diophantine quadruple in the sense of Szalay and Ziegler for the set $$\mathcal{S} $$ of integers composed only of primes p and p + 2.

Diagonal genus 5 curves, elliptic curves over $\mathbb {Q}(t)$, and rational diophantine quintuples

The problem of finding all possible extensions of a given rational diophantine quadruple to a rational diophantine quintuple is equivalent to the determination of the set of rational points on a certain curve of genus 5 that can be written as an intersection of three diagonal quadrics in $\mathbb P^4$. We discuss how one can (try to) determine the set of rational points on such a curve. We apply our approach to the original question in several cases. In particular, we show that Fermat's diophantine quadruple (1,3,8,120) can be extended to a rational diophantine quintuple in only one way, namely by 777480/8288641. We then discuss a method that allows us to find the Mordell-Weil group of an elliptic curve $E$ defined over the rational function field $\mathbb Q(t)$ when $E$ has full $\mathbb Q(t)$-rational 2-torsion. This builds on recent results of Dujella, Gusi\'c and Tadi\'c. We give several concrete examples to which this method can be applied. One of these results implies that there is only one extension of the diophantine quadruple $\bigl(t-1,t+1,4t,4t(4t^2-1)\bigr)$ over $\mathbb Q(t)$.

There is no Diophantine quintuple

A set of m m positive integers { a 1 , a 2 , … , a m } \{a_1, a_2, \ldots , a_m\} is called a Diophantine m m -tuple if a i a j + 1 a_i a_j + 1 is a perfect square for all 1 ≤ i > j ≤ m 1 \le i > j \le m . Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine m m -tuples states that no Diophantine quintuple exists at all. We prove this conjecture.

A Pellian Equation with Primes and Applications to $$D(-1)$$ D ( - 1 ) -Quadruples

In this paper, we prove that the equation $x^2-(p^{2k+2}+1)y^2=-p^{2l+1}$, $l \in \{0,1,\dots,k\}, k \geq 0$, where $p$ is an odd prime number, is not solvable in positive integers $x$ and $y$. By combining that result with other known results on the existence of Diophantine quadruples, we are able to prove results on the extensibility of some $D(-1)$-pairs to quadruples in the ring $\mathbb{Z}[\sqrt{-t}], t>0$.

On the number of extensions of a Diophantine triple

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight.

There are no Diophantine quadruples of Fibonacci numbers

There are no Diophantine quadruples of Fibonacci numbers

The regularity of Diophantine quadruples

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by the unity is a perfect square. A conjecture on the regularity of Diophantine quadruples asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element of the triple. The problem is reduced to studying an equation expressed as the coincidence of two linear recurrence sequences with initial terms composed of the fundamental solutions of some Pellian equations. In this paper, we determine the values of those initial terms completely and obtain finiteness results on the number of solutions of the equation. As one of the applications to the problem on the regularity of Diophantine quadruples, we show in general that the number of ways of extending any given Diophantine triple is at most $11$.

Diophantine quadruples of numbers whose elements are in proportion

In this paper certain non-F-type $P_{3,k}$ sequences which contain Diophantine quadruples of numbers in proportion are presented. It is proved that there exist an infinite number of non-F-type $P_{3,k}$ sequences which possess Diophantine quadruples of numbers in proportion.

Another generalization of a theorem of Baker and Davenport

Dujella and Pethő, generalizing a result of Baker and Davenport, proved that the set {1,3} cannot be extended to a Diophantine quintuple. As a consequence of our main result, we show that the Diophantine pair {1,b} is regular if b−1 is a prime power.

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.

Diophantine quintuples containing triples of the first kind

We consider Diophantine quintuples \(\{a, b, c, d, e\}\), sets of integers with \(a<b<c<d<e\) the product of any two elements of which is one less than a perfect square. Triples of the first kind are sets \(\{A, B, C\}\) with \(C\ge B^{5}\). We show that there are no Diophantine quintuples \(\{a, b, c, d, e\}\) such that \(\{a, b, d\}\) is a triple of the first kind.

Searching for Diophantine quintuples

We consider Diophantine quintuples $\{a, b, c, d, e\}$. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current esti

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}; ; .

Bounds on the number of Diophantine quintuples

We consider Diophantine quintuples {a,b,c,d,e}. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most 2.3⋅1029 Diophantine quintuples.

Further remarks on Diophantine quintuples

A set of $m$ positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a <i>Diophantine $m$-tuple</i>. Much work has been done attempting to prove that there exist no Diophantine quintuples. I

On the prime divisors of elements of a D(-1) quadruple

We show that if {1, b, c, d} is a D(-1) diophantine quadruple with b<c<d and c=1+s^2, then the cases s=p^k, s=2p^k, c=p and c=2p^k do not occur, where p is an odd prime and k is a positive integer. For the integer d=1+x^2, we show that it is not prime and that x is divisible by at least two distinct odd primes. Furthermore, we present several infinite families of integers b such that the D(-1) pair {1, b} cannot be extended to a D(-1) quadruple. For instance, we show that if r=5p where p is an odd prime, then the D(-1) pair {1, r^2+1} cannot be extended to a D(-1) quadruple.

The extendibility of Diophantine pairs II: Examples

This paper is a continuation of a recent paper (see [10]), in which for a fixed Diophantine pair {a,b} with a<b, we gave an upper bound for minimal c such that {a,b,c,d} is an irregular Diophantine quadruple with b<c<d. Here, we provide three very good examples to support the theory developed in [10].

On Diophantine quintuple conjecture

In this note, we prove that if $\{a,b,c,d,e\}$ with $a<b<c<d<e$ is a Diophantine quintuple, then $d<10^{76}$.

On Diophantine quintuples and $$D(-1)$$ D ( - 1 ) -quadruples

In this paper the known upper bound $$10^{96}$$ for the number of Diophantine quintuples is reduced to $$6.8\cdot 10^{32}$$ . The key ingredient for the improvement is that certain individual bounds on parameters are now combined with a more efficient counting of tuples, and estimated by sums over divisor functions. As a side effect, we also improve the known upper bound $$4\cdot 10^{70}$$ for the number of $$D(-1)$$ -quadruples to $$5\cdot 10^{60}$$ .

-quadruples in the ring , for some exceptional cases of z

Abstract By the work of Abu Muriefah, Al-Rashed, Dujella and Soldo, the problem of the existence of D ( z ) -quadruples in the ring Z [ − 2 ] has been solved, except for the cases z = 24 a + 2 + ( 12 b + 6 ) − 2 , z = 24 a + 5 + ( 12 b + 6 ) − 2 , z = 48 a + 44 + ( 24 b + 12 ) − 2 . We present some new formulas for D ( z ) -quadruples in these remaining cases, involving some congruence conditions modulo 11 on integers a and b. We show the existence of D ( z ) -quadruple for significant proportion of the remaining three cases. These new formulas then together with previous results on this subject allow us to almost completely characterize elements z of Z [ − 2 ] for which a Diophantine quadruple with the property D ( z ) exists.