Diophantine Representation of Fibonacci Numbers Over Natural Numbers

1990 ◽  
pp. 197-201 ◽  
Author(s):  
James P. Jones
Keyword(s):  
Fibonacci Numbers ◽  
Natural Numbers ◽  
Diophantine Representation

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.

Representing Numbers Using Fibonacci Variants

2015 ◽  
Author(s):  
Stephen K. Lucas
Keyword(s):  
Natural Number ◽  
Greedy Algorithm ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Fixed Number ◽  
Natural Numbers ◽  
Form Versus ◽  
Entire Number

This chapter introduces the Zeckendorf representation of a Fibonacci sequence, a form of a natural number which can be easily found using a greedy algorithm: given a number, subtract the largest Fibonacci number less than or equal to it, and repeat until the entire number is used up. This chapter first compares the efficiency of representing numbers using Zeckendorf form versus traditional binary with a fixed number of digits and shows when Zeckendorf form is to be preferred. It also shows what happens when variants of Zeckendorf form are used. Not only can natural numbers as be presented sums of Fibonacci numbers, but arithmetic can also be done with them directly in Zeckendorf form. The chapter includes a survey of past approaches to Zeckendorf representation arithmetic, as well as some improvements.

POSSIBILITIES AND IMPOSSIBILITIES IN SQUARE-TILING

2011 ◽  
Vol 21 (05) ◽  
pp. 545-558
Author(s):  
A. M. BERKOFF ◽  
J. M. HENLE ◽  
A. E. MCDONOUGH ◽  
A. P. WESOLOWSKI
Keyword(s):  
Fibonacci Numbers ◽  
Natural Numbers

A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of sidelength n for every n in the set. From Ref. 8 we know that ℕ itself tiles the plane. From that and Ref. 9 we know that the set of even numbers tiles the plane while the set of odd numbers doesn't. In this paper we explore the nature of this property. We show, for example, that neither tiling nor non-tiling is preserved by superset. We show that a set with one or three odd numbers may tile the plane—but a set with two odd numbers can't. We find examples of both tiling and non-tiling sets that can be partitioned into tiling sets, non-tiling sets or a combination. We show that any set growing faster than the Fibonacci numbers cannot tile the plane.

THE FIBONACCI-NORM OF A POSITIVE INTEGER: OBSERVATIONS AND CONJECTURES

2010 ◽  
Vol 06 (02) ◽  
pp. 371-385 ◽  
Author(s):  
JEONG SOON HAN ◽  
HEE SIK KIM ◽  
J. NEGGERS
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Natural Numbers

In this paper, we define the Fibonacci-norm [Formula: see text] of a natural number n to be the smallest integer k such that n|Fk, the kth Fibonacci number. We show that [Formula: see text] for m ≥ 5. Thus by analogy we say that a natural number n ≥ 5 is a local-Fibonacci-number whenever [Formula: see text]. We offer several conjectures concerning the distribution of local-Fibonacci-numbers. We show that [Formula: see text], where [Formula: see text] provided Fm+k ≡ Fm (mod n) for all natural numbers m, with k ≥ 1 the smallest natural number for which this is true.

The Argument from (Natural) Numbers

2018 ◽  
Author(s):  
Tyron Goldschmidt
Keyword(s):  
Natural Numbers ◽  
Existence Of God ◽  
Divine Attribute ◽  
Rule Out

This chapter considers Plantinga’s argument from numbers for the existence of God. Plantinga sees divine psychologism as having advantages over both human psychologism and Platonism. The chapter begins with Plantinga’s description of the argument, including the relation of numbers to any divine attribute. It then argues that human psychologism can be ruled out completely. However, what rules it out might rule out divine psychologism too. It also argues that the main problem with Platonism might also be a problem with divine psychologism. However, it will, at the least, be less of a problem. In any case, there are alternative, possibly viable views about the nature of numbers that have not been touched by Plantinga’s argument. In addition, the chapter touches on the argument from properties, and its relation to the argument from numbers.

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