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.

Semi-local convergence of a Newton-like method for solving equations with a singular derivative

2018 ◽  
Vol 27 (1) ◽  
pp. 01-08
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
GEORGE SANTHOSH ◽  
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Real Line ◽  
Local Convergence ◽  
Approximate Solutions ◽  
Convergence Domain ◽  
Solving Equations ◽  
Solutions Of Equations ◽  
Local Convergence Analysis ◽  
Special Case

We present a semi-local convergence analysis for a Newton-like method to approximate solutions of equations when the derivative is not necessarily non-singular in a Banach space setting. In the special case when the equation is defined on the real line the convergence domain is improved for this method when compared to earlier results. Numerical results where earlier results cannot apply but the new results can apply to solve nonlinear equations are also presented in this study.

Order of operations in elementary arithmetic

1962 ◽  
Vol 9 (5) ◽  
pp. 263-267
Author(s):  
Marvin L. Bender
Keyword(s):  
Natural Numbers ◽  
Whole Numbers ◽  
Restricted Sense

We will assume acquaintance with the set of natural numbers N = {1, 2, 3, …} and the set of whole numbers W = {0, 1, 2, 3, …}. The fundamental operations of addition and multiplication, and the related operations of subtraction and division will also be assumed as well known to the reader. It is to be understood that subtraction and division are operations only in a restricted sense, since it is not always possible to subtract or divide in the set of whole numbers. A recognition of a few of the properties of these operations, at least in form if not in name, will be assumed: in particular, the commutative and associative properties of addition and multiplication, and the fact that they do not hold for subtraction and division. Finally, an understanding of the meaning of the relations indicated by =(equal) and ≠ (not equal) will be assumed.

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.

scholarly journals The Properties of the Arithmetic Function as the Consecutive Sum of Digits in Natural Numbers

2021 ◽  
Author(s):  
Ramazanali Maleki Chorei
Keyword(s):  
Natural Number ◽  
Diophantine Equations ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Natural Numbers ◽  
Whole Numbers ◽  
Sum Of Digits ◽  
Dimensional Equation

In this paper defines the consecutive sum of the digits of a natural number, so far as it becomes less than ten, as an arithmetic function called and then introduces some important properties of this function by proving a few theorems in a way that they can be used as a powerful tool in many cases. As an instance, by introducing a test called test, it has been shown that we are able to examine many algebraic equalities in the form of in which and are arithmetic functions and to easily study many of the algebraic and diophantine equations in the domain of whole numbers. The importance of test for algebraic equalities can be considered equivalent to dimensional equation in physics relations and formulas. Additionally, this arithmetic function can also be useful in factorizing the composite odd numbers.

scholarly journals DIAGNOSTICS OF MATHEMATICAL PROOF OF THE BEAL CONJECTURE IN MEDICAL PSYCHOLOGY (REMAKE OF PREVIOUS AUTHOR’S ARTICLES CONCERNING FERMAT’S LAST THEOREM)

2021 ◽  
Vol 1 (5(69)) ◽  
pp. 28-33
Author(s):  
Y. Ivliev
Keyword(s):  
Mathematical Proof ◽  
Mathematical Structure ◽  
Pythagorean Theorem ◽  
Medical Psychology ◽  
Natural Numbers ◽  
Whole Numbers ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
The Given ◽  
Infinite Descent

In the given work diagnostics of mathematical proof of the Beal Conjecture (Generalized Fermat’s Last Theorem) obtained in the earlier author’s works was conducted and truthfulness of the suggested proof was established. Realizing the process of the Bill Conjecture solution, the mathematical structure defining hypothetical equality of the Fermat theorem was determined. Such a structure turned to be one of Pythagorean theorem with whole numbers. With help of Euclid’s geometrical theorem and Fermat’s method of infinite descent one can manage to set that Pythagorean equation in whole numbers representing Fermat’s Last Theorem cannot exist and then the Fermat theorem is true, that is Fermat’s equality in natural numbers does not exist. Thus mental scheme of “demonstratio mirabile”, which Pierre de Fermat mentioned on the margins of Diophantus’s “Arithmetic”, was reconstructed. 

Discovering properties of the natural numbers

1965 ◽  
Vol 12 (8) ◽  
pp. 627-632
Author(s):  
Harriet Griffin
Keyword(s):  
Elementary School ◽  
Natural Numbers ◽  
Whole Numbers ◽  
Mathematical Properties ◽  
Mathematical Rule ◽  
Insight Into

Have you ever known the joy of discovering a mathematical rule or relation? Surely you can recall the feeling of satisfaction that comes from finding a proof of a theorem, whether or not you formulated its statement. In this paper we propose to review some of the basic rules about t he natural numbers 1, 2, 3, etc., often called the whole numbers or arithmetic integers, to show that these numbers provide the superior student in the upper grades of the elementary school with many opportunities for discerning mathematical properties. Such considerations will not only sharpen the student's insight into like problems, but will also help him to develop a relish for such thoughtful activity. Investigations he can make independently, after some fundamental ideas have been explained, will also convince him of the need for distinguishing between the process of examining objects to draft a statement about a characteristic they possess and the task of constructing a proof to show whether or not the statement is correct.

Aristotle on Species Variation

Philosophy ◽  
1986 ◽  
Vol 61 (236) ◽  
pp. 245-252 ◽  
Author(s):  
James Franklin
Keyword(s):  
Infinite Number ◽  
Chemical Elements ◽  
The Other ◽  
Species Variation ◽  
Whole Numbers ◽  
Other Hand ◽  
Over Time

The whole numbers and the chemical elements vary discretely: 5 is the next number to 4 and there is no number between them; silver is next to gold in the atomic table and there is no element between them. On the other hand, colours vary continuously: between red and yellow there is another colour, orange, between orange and yellow there is another colour, and so on. Between any two colours, no matter how close, there is an intermediate colour–indeed, an infinite number of intermediate colours. A surface may change gradually over time from red to yellow, assuming all the colours in between. Or again, a surface may be red at one edge and yellow at the other, changing gradually (over space) and assuming all the colours in between.

Non-constructive rules of inference

2018 ◽  
Author(s):  
A.P. Hazen
Keyword(s):  
Natural Number ◽  
Infinite Number ◽  
Deductive System ◽  
Higher Order ◽  
The Other ◽  
Natural Numbers ◽  
Substitutional Quantification ◽  
Deductive Systems ◽  
Formal Systems ◽  
Order Arithmetic

For some theoretical purposes, generalized deductive systems (or, ‘semi-formal’ systems) are considered, having rules with an infinite number of premises. The best-known of these rules is the ‘ω-rule’, or rule of infinite induction. This rule allows the inference of ∀nΦ(n) from the infinitely many premises Φ(0), Φ(1),… that result from replacing the numerical variable n in Φ(n) with the numeral for each natural number. About 1930, in part as a response to Gödel’s demonstration that no formal deductive system had as theorems all and only the true formulas of arithmetic, several writers (most notably, Carnap) suggested considering the semi-formal systems obtained, from some formulation of arithmetic, by adding this rule. Since no finite notation can provide terms for all sets of natural numbers, no comparable rule can be formulated for higher-order arithmetic. In effect, the ω-rule is valid just in case the relevant quantifier can be interpreted substitutionally; looked at from the other side, the validity of some analogue of the ω-rule is the essential mathematical characteristic of substitutional quantification.

DISTRIBUTION OF PRIME AND COMPOSITE NUMBERS AND THEIR ALGORHYTHMIC APPENDICES

2017 ◽  
pp. 48-64 ◽  
Author(s):  
Sergey Ivanovich Chermidov
Keyword(s):  
Infinite Number ◽  
Diophantine Equations ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Independent Verification ◽  
Composite Number ◽  
Verification Test

The article focuses on methods defining and distributing the composite numbers, prime numbers, twins of prime numbers and composite numbers of twins that do not have divisors 2 and 3 in N , based on the set of numbers of type Θ = {6 k ± 1 / kN } where N is the set of all natural numbers, which is a semigroup with respect to multiplication. The calculation the exact quantity of primes in a given interval is given. A method for obtaining prime numbers p ≥ 5 by their ordinal numbers in a set of primes p ≥ 5 is proposed, as well as a new algorithm for finding and distributing prime numbers on the basis of the closeness of the set Θ. The article shows that any composite number n Θ is representable as products (6 x ± 1) (6 y ± 1), where x, yN are the natural solutions of one of the four Diophantine equations P ( x , y , λ) = 0 : 6 × xy ± x ± y - λ = 0. It has been proved that if there is a parameter λ of twins of prime numbers, then none of the Diophantine P ( x , y , λ) = 0 equations has any solutions. A new universal, deterministic, polynomial and independent verification test is provided for the simplicity of the numbers of a species 6 × k ± 1. Algorithms of distributions of parameters of twins of prime numbers and parameters composite numbers of twins are given, they are not divisible by 2 and 3, and variants of proofs for their infinite number are given.

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