On the Diophantine pair {a,3a}

Adédji, Kouèssi Norbert; He, Bing; Togbé, Alain

doi:10.1016/j.jnt.2021.03.011

Cited by 3 publications

(3 citation statements)

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“…We follow the strategy used in [2] with some improvements and define the following linear forms in three logarithms (2.17) Λ = l log α − m log β + log γ for c > b, and…”

Section: 2mentioning

confidence: 99%

“…In this paper, we consider extensions of the D(4)-pairs {a, ka}, k = 2, 3, 6 to a D(4)-quadruple {a, ka, c, d} by following the method in [2], where the extensions of Diophantine pairs {a, ka}, with k = 3, 8, were studied. We conjecture that d = d + (a, ka, c), i.e., that there is no irregular quadruple of this form.…”

Section: Introductionmentioning

confidence: 99%

“…The theory of Pellian equations guarantees that there is only one fundamental solution (r 1 , a 1 ) of (1.2), for any k = 2, 3, 6, namely (r 1 , a 1 ) ∈ { (6,4), (4,2), (10,4)} (in that order). All solutions (r p , a p ) of the equation (1.2) are given by…”

Section: Introductionmentioning

confidence: 99%

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On the $D(4)$-pairs $\{a, ka\}$ with $k\in \{2,3,6\}$

Adédji¹,

Trebješanin²,

Filipin³

et al. 2023

Glas. Mat. Ser. III

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Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that $ab+4$ is a perfect square. In this paper, we study the extensibility of the $D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering families of positive integers $c$ depending on $a,$ if $\{a, b, c, d\}$ is a set of positive integers which has the property that the product of any two of its elements increased by $4$ is a perfect square, then $d$ is given by d=a+b+c+1/2(abc±√((ab+4)(ac+4)(bc+4))). As a corollary, we prove that any $D(4)$-quadruple tht contains the pair $\{a, ka\}$ is regular.

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“…We follow the strategy used in [2] with some improvements and define the following linear forms in three logarithms (2.17) Λ = l log α − m log β + log γ for c > b, and…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%