Effective solution of the D(-1)-quadruple conjecture

Dujella, Andrej; Filipin, Alan; Fuchs, Clemens

doi:10.4064/aa128-4-2

Cited by 36 publications

(41 citation statements)

References 22 publications

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“…In the case of n = −1, there are several results supporting the validity of this conjecture ( [1,9,14,15,19,20]), and it has been almost completely settled. More precisely, there does not exist a D(−1)-quintuple ( [15]) and there exist only finitely many D(−1)-quadruples ( [14]).…”

Section: Conjecture 12 ([7]mentioning

confidence: 81%

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The non-extensibility of D(4k)-triples {1, 4k(k-1), 4k^2+1} with |k| prime

Fujita¹

2006

Glas. Mat. Ser. III

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Section: Conjecture 12 ([7]mentioning

confidence: 81%

“…More precisely, there does not exist a D(−1)-quintuple ( [15]) and there exist only finitely many D(−1)-quadruples ( [14]). Suppose now that there exists a D(−4)-quadruple {a 1 , a 2 , a 3 , a 4 }.…”

Section: Conjecture 12 ([7]mentioning

confidence: 99%

The non-extensibility of D(4k)-triples {1, 4k(k-1), 4k^2+1} with |k| prime

Fujita¹

2006

Glas. Mat. Ser. III

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“…We have not been able to find neither D(1)-set nor D(−1)-set {a, b, c} which is also a D(n)-set for four distinct n's with |n| > 1, and in fact we suspect that |n 1 (5)| > 1 based on exhaustive but unsuccessful computer search. (We remark that it has been conjectured that there do not exist D(−1)-quadruples, and that it is known that there do not exist D(−1)-quintuples and that there are at most finitely many D(−1)-quadruples [9,10]). …”

Section: A Modification Of the Problemmentioning

confidence: 86%

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

Adžaga

Dujella

Kreso

et al. 2018

Journal of Number Theory

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For a nonzero integer n, a set of distinct nonzero integers {a 1 , a 2 , . . . , a m } such that a i a j + 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 as simply 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.

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“…(For the elements of the set S, it is still not known if such a quadruple exists. Recently, Dujella, Filipin and Fuchs [7] proved that there exist only finitely many D(−1)-quadruples and D(−4)-quadruples.) In the ring of Gaussian integers Z[i], the analogous statement can be proved (see [6]).…”

Section: Introductionmentioning

confidence: 98%