1. Diophantine quadruples and near-Diophantine quintuples from P3,K sequences
- Author
-
A. M. S. Ramasamy
- Subjects
Discrete mathematics ,Sequence ,Polynomial ,Fibonacci number ,Diophantine set ,General Mathematics ,Diophantine equation ,010102 general mathematics ,Natural number ,01 natural sciences ,Square (algebra) ,010101 applied mathematics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Pell's equation ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
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.
- Published
- 2017