Diophantine quadruples containing some triples and the number of Diophantine quintuples

2008 ◽  
Author(s):  
Yasutsugu Fujita ◽  
Takao Komatsu
Keyword(s):  
Diophantine Quintuples ◽  
Diophantine Quadruples

Diophantine quadruples and near-Diophantine quintuples from P3,K sequences

2017 ◽  
Vol 10 (01) ◽  
pp. 1750010
Author(s):  
A. M. S. Ramasamy
Keyword(s):  
Infinite Number ◽  
Fibonacci Numbers ◽  
Natural Numbers ◽  
Text Type ◽  
Type Formula ◽  
Fourth Term ◽  
Unexplored Area ◽  
Diophantine Quintuples ◽  
Diophantine Quadruples

The question of a non-[Formula: see text]-type [Formula: see text] sequence wherein the fourth term shares the property [Formula: see text] with the first term has not been investigated so far. The present paper seeks to fill up the gap in this unexplored area. Let [Formula: see text] denote the set of all natural numbers and [Formula: see text] the sequence of Fibonacci numbers. Choose two integers [Formula: see text] and [Formula: see text] with [Formula: see text] such that their product increased by [Formula: see text] is a square [Formula: see text]. Certain properties of the sequence [Formula: see text] defined by the relation [Formula: see text] are established in this paper and polynomial expressions for Diophantine quadruples from the [Formula: see text] sequence [Formula: see text] are derived. The concept of a near-Diophantine quintuple is introduced and it is proved that there exist an infinite number of near-Diophantine quintuples.

The regularity of Diophantine quadruples

2017 ◽  
Vol 370 (6) ◽  
pp. 3803-3831 ◽  
Author(s):  
Yasutsugu Fujita ◽  
Takafumi Miyazaki
Keyword(s):  
Diophantine Quadruples

scholarly journals Adjugates of Diophantine Quadruples

Integers ◽  
2010 ◽  
Vol 10 (2) ◽  
Author(s):  
Philip Gibbs
Keyword(s):  
Diophantine Quadruples

AbstractDiophantine

scholarly journals ON THE FAMILY OF DIOPHANTINE TRIPLES {k− 1,k+ 1, 16k3− 4k}

2007 ◽  
Vol 49 (2) ◽  
pp. 333-344 ◽  
Author(s):  
YANN BUGEAUD ◽  
ANDREJ DUJELLA ◽  
MAURICE MIGNOTTE
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
The Family ◽  
Diophantine Triples ◽  
Image Position ◽  
Diophantine Quadruples

AbstractIt is proven that ifk≥ 2 is an integer anddis a positive integer such that the product of any two distinct elements of the setincreased by 1 is a perfect square, thend= 4kord= 64k5−48k3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k− 1,k+ 1,c,d} are regular.

scholarly journals Diophantine quintuples containing triples of the first kind

2016 ◽  
Vol 72 (2) ◽  
pp. 235-242
Author(s):  
David J. Platt ◽  
Timothy S. Trudgian
Keyword(s):  
Diophantine Quintuples

Diophantine quadruples in $\mathbb {Z}[\sqrt {4k+3}]$

2007 ◽  
Vol 17 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Zrinka Franušić
Keyword(s):  
Diophantine Quadruples

scholarly journals The extensibility of the Diophantine triple {2, b, c}

2021 ◽  
Vol 29 (2) ◽  
pp. 5-24
Author(s):  
Nikola Adžaga ◽  
Alan Filipin ◽  
Ana Jurasić
Keyword(s):  
Diophantine Triple ◽  
Diophantine Quadruples

Abstract The aim of this paper is to consider the extensibility of the Diophantine triple {2, b, c}, where 2 < b < c, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of c’s (depending on b). As corollary, for example, we prove that for b/2 − 1 prime, all Diophantine quadruples {2, b, c, d} with 2 < b < c < d are regular.

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