The number of Diophantine quintuples II

2013 ◽  
Vol 82 (2) ◽  
pp. 293-308 ◽  
Author(s):  
ALAN FILIPIN ◽  
YASUTSUGU FUJITA
Keyword(s):  
Diophantine Quintuples

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 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.

Bounds for Diophantine quintuples II

2016 ◽  
Vol 88 (1-2) ◽  
pp. 59-78
Author(s):  
MIHAI CIPU ◽  
ALAN FILIPIN ◽  
YASUTSUGU FUJITA
Keyword(s):  
Diophantine Quintuples

scholarly journals On the size of Diophantine m-tuples

2002 ◽  
Vol 132 (1) ◽  
pp. 23-33 ◽  
Author(s):  
ANDREJ DUJELLA
Keyword(s):  
The Other ◽  
Famous Conjecture ◽  
Nonzero Integer ◽  
Other Hand ◽  
Parametric Families ◽  
Diophantine Quintuples ◽  
Positive Integers

Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D(1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [12, 14, 16], but the conjecture is still unproved.On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920, 32768, 44460, 19534284} with the property D(2985984) [19]].

scholarly journals On Diophantine quintuples

1997 ◽  
Vol 81 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Andrej Dujella
Keyword(s):  
Diophantine Quintuples

scholarly journals The number of Diophantine quintuples

2010 ◽  
Vol 45 (1) ◽  
pp. 15-29 ◽  
Author(s):  
Yasutsugu Fujita
Keyword(s):  
Diophantine Quintuples
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