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.

Some varieties of algebraic systems of type ((n),(m))

2019 ◽  
Vol 12 (01) ◽  
pp. 1950005
Author(s):  
D. Phusanga ◽  
J. Koppitz
Keyword(s):  
Algebraic System ◽  
Natural Numbers ◽  
Algebraic Systems ◽  
Term Operation ◽  
Type Formula

In the present paper, we classify varieties of algebraic systems of the type [Formula: see text], for natural numbers [Formula: see text] and [Formula: see text], which are closed under particular derived algebraic systems. If we replace in an algebraic system the [Formula: see text]-ary operation by an [Formula: see text]-ary term operation and the [Formula: see text]-ary relation by the [Formula: see text]-ary relation generated by an [Formula: see text]-ary formula, we obtain a new algebraic system of the same type, which we call derived algebraic system. We shall restrict the replacement to so-called “linear” terms and atomic “linear” formulas, respectively.

scholarly journals On Diophantine quadruples of Fibonacci numbers

2017 ◽  
Vol 52 (2) ◽  
pp. 221-234
Author(s):  
Yasutsugu Fujita ◽  
◽  
Florian Luca ◽  
Keyword(s):  
Fibonacci Numbers ◽  
Diophantine Quadruples

scholarly journals Numerical semigroups of small and large type

2021 ◽  
pp. 1-20
Author(s):  
Deepesh Singhal
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Natural Numbers ◽  
Type Formula ◽  
Explicit Characterization ◽  
Large Type ◽  
Number Formula ◽  
Genus Formula

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].

scholarly journals There are no Diophantine quadruples of Fibonacci numbers

2018 ◽  
Vol 185 (1) ◽  
pp. 19-38
Author(s):  
Yasutsugu Fujita ◽  
Florian Luca
Keyword(s):  
Fibonacci Numbers ◽  
Diophantine Quadruples

scholarly journals The Methods of Solving Equations Ax+By=Cz with Co-Prime A, B, C, where x≥2,y≥2,x≥2 Are Natural Numbers, Equal the Two only in one of the Three Possible Cases - The Proof of Catalan's Conjecture

2013 ◽  
Vol 8 ◽  
pp. 17-25
Author(s):  
K. Raja Rama Gandhi ◽  
Reuven Tint ◽  
Michael Tint
Keyword(s):  
Infinite Number ◽  
Natural Numbers ◽  
Whole Numbers ◽  
Solving Equations ◽  
Solutions Of Equations

One of the principal problems of the Beal's conjecture, as we see that, is methods for finding a pairwise coprime solution which is defined below. First found methods and identities, allowing receiving infinite number solutions of equations as Ax+By=Cz for co-prime integers arranged in a pair (A,B,C)=1 are natural (whole) numbers, where a fixed permutation (x,y,z)corresponds to each of the permutations (2,3,4), (2,4,3), (4,3,2) Here we obtain also our method and identities of all not recurrent and not co-prime solutions of the above type, part of which has already been published, in contrast to the method of obtaining the recurrence not co-prime solutions of this type from [(1), W. Sierpiński, p. 21-25, 63]. As the solution of the main problem appeared additional problems that solved by obtained appropriate identities. Given as two equal proofs of Catalan's Conjecture.

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