scholarly journals RP-181: Formulation of Standard Quadratic Congruence of Even Composite Modulus modulo an Even Prime Raised to the Power n

2021 ◽  
Vol 9 (8) ◽  
pp. 3064-3069
Author(s):  
Prof B M Roy
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Complete Formulation ◽  
Composite Modulus ◽  
Quadratic Congruence ◽  
Keywords Composite

Abstract: The paper presented here, is a standard quadratic congruence of composite modulus, studied rigorously and found the formulation incomplete. It was partially formulated by the earlier mathematicians. The present authors realised that the earlier formulation need a completion and a reformulation of the solutions is done along with two more results. The author considered the problem for reformulation, studied and reformulated the solutions completely. A partial formulation is found in a books of Number Theory by Zuckerman at el. There the formulation is only for an odd positive integer but nothing is said about even positive integer. The authors have provided a complete formulation of the said quadratic congruence and presented here. Keywords: Composite Modulus, Quadratic Congruence, Reformulation.

scholarly journals RP-179: Formulation of Solutions of Standard Bi-Quadratic Congruence Modulo a POSITIVE INTEGER MULTIPLE to nth Power of Four

2021 ◽  
Vol 9 (VII) ◽  
pp. 2598-2601
Author(s):  
Prof. B. M. Roy
Keyword(s):  
Positive Integer ◽  
Integer Multiple ◽  
Composite Modulus ◽  
First Case ◽  
Congruence Modulo ◽  
Quadratic Congruence ◽  
First Time

In this paper, the author has formulated the solutions of the standard bi-quadratic congruence of an even composite modulus modulo a positive integer multiple to nth power of four. First time a formula is established for the solutions. No literature is available for the current congruence. The author analysed the formulation of solutions in two different cases. In the first case of analysis, the congruence has the formulation which gives exactly eight incongruence solutions while in the second case of the analysis, the congruence has a different formulation of solutions and gives thirty-two incongruent solutions. A very simple and easy formulation to find all the solutions is presented here. Formulation is the merit of the paper.

A SURVEY OF FACTORIZATION COUNTING FUNCTIONS

2005 ◽  
Vol 01 (04) ◽  
pp. 563-581 ◽  
Author(s):  
A. KNOPFMACHER ◽  
M. E. MAYS
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Additive Number Theory ◽  
Recursive Algorithms ◽  
Additive Number ◽  
Combinatorial Objects ◽  
Survey Techniques ◽  
Counting Functions ◽  
General Field ◽  
Positive Integers

The general field of additive number theory considers questions concerning representations of a given positive integer n as a sum of other integers. In particular, partitions treat the sums as unordered combinatorial objects, and compositions treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distinct, or relatively prime, or all congruent to ±1 modulo 5. In this paper we review work on analogous problems concerning representations of n as a product of positive integers. We survey techniques for enumerating product representations both in the unrestricted case and in the case when the factors are required to be distinct, and both when the product representations are considered as ordered objects and when they are unordered. We offer some new identities and observations for these and related counting functions and derive some new recursive algorithms to generate lists of factorizations with restrictions of various types.

scholarly journals A Ramsey-type property in additive number theory

1985 ◽  
Vol 27 ◽  
pp. 5-10
Author(s):  
S. A. Burr ◽  
P. Erdös
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Additive Number Theory ◽  
Additive Number ◽  
Large Positive Integer ◽  
Ramsey Type ◽  
Type Property ◽  
Positive Integers

Let A be a sequence of positive integers. Define P(A) to be the set of all integers representable as a sum of distinct terms of A. Note that if A contains a repeated value, we are free to use it as many times as it occurs in A. We call A complete if every sufficiently large positive integer is in P(A), and entirely complete if every positive integer is in P(A). Completeness properties have received considerable, if somewhat sporadic, attention over the years. See Chapter 6 of [3] for a survey.

scholarly journals On the sum of digits of some sequences of integers

2013 ◽  
Vol 11 (1) ◽  
Author(s):  
Javier Cilleruelo ◽  
Florian Luca ◽  
Juanjo Rué ◽  
Ana Zumalacárregui
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Integer Sequences ◽  
Sum Of Digits ◽  
Fixed Positive Integer ◽  
Almost All

AbstractLet b ≥ 2 be a fixed positive integer. We show for a wide variety of sequences {a n}n=1∞ that for almost all n the sum of digits of a n in base b is at least c b log n, where c b is a constant depending on b and on the sequence. Our approach covers several integer sequences arising from number theory and combinatorics.

Number theory and other reminiscences of Viscount Cherwell

1988 ◽  
Vol 42 (2) ◽  
pp. 197-204 ◽  
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Fundamental Theorem ◽  
Prime Numbers ◽  
Christ Church ◽  
Active Interest ◽  
Fundamental Theorem Of Arithmetic ◽  
Elegant Proof ◽  
Theory Of Numbers

Lord Cherwell (i) was, of course, a very distinguished ex-perimental physicist but he had (like many others) a considerable active interest in the theory of numbers. I met him in 1930 when Christ Church, Oxford, elected me to a Senior (postgraduate) Scholarship and I migrated there from my original college. Cherwell’s first published work (2) in the theory of numbers was a very simple and elegant proof of the fundamental theorem of arithmetic, that any positive integer can be expressed as a product of prime numbers in just one way (apart from a possible rearrangement of the order of the factors). (A prime is a positive integer greater than 1 whose only factors are 1 and itself.) His proof is by the method of descent (used by Fermat, but not for this problem). Assume the fundamental theorem false and call any number that can be expressed as a product of primes in two or more ways abnormal.

scholarly journals Elementary methods in the theory of numbers

1939 ◽  
Vol 31 ◽  
pp. xvi-xxiii
Author(s):  
S. A. Scott
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Exponential Function ◽  
Monotone Sequence ◽  
Binomial Theorem ◽  
Irrational Numbers ◽  
Early Stages ◽  
Elementary Methods ◽  
Usual Notation ◽  
Theory Of Numbers

§ 1. The importance of proving inequalities of an essentially algebraic nature by “elementary” methods has been emphasised by Hardy (Prolegomena to a Chapter on Inequalities), and by Hardy, Littlewood and Polya (Inequalities). The object of this Note is to show how some of the results in the early stages of Number Theory can be obtained by making a minimum appeal to irrational numbers and the notion of a limit. We use the elementary notion of a logarithm to a base “a” > 1, and make no appeal to the exponential function. The Binomial Theorem is only used for a positive integer index. Our minimum appeal rests in the assumption that a bounded monotone sequence tends to a limit. We adopt throughout the usual notation. Finally, it need scarcely be added that the methods employed are not claimed to be new.

On the addition of two weighted squares of units mod n

2016 ◽  
Vol 12 (07) ◽  
pp. 1783-1790 ◽  
Author(s):  
Cui-Fang Sun ◽  
Zhi Cheng
Keyword(s):  
Positive Integer ◽  
Number Of Solutions ◽  
Residue Classes ◽  
Quadratic Congruence

For any positive integer [Formula: see text], let [Formula: see text] be the ring of residue classes modulo [Formula: see text] and [Formula: see text] be the group of its units. Recently, for any [Formula: see text], Yang and Tang obtained a formula for the number of solutions of the quadratic congruence [Formula: see text] with [Formula: see text] units, nonunits and mixed pairs, respectively. In this paper, for any [Formula: see text], we give a formula for the number of representations of [Formula: see text] as the sum of two weighted squares of units modulo [Formula: see text]. We resolve a problem recently posed by Yang and Tang.

scholarly journals Elementary Number Theory Problems. Part II

2021 ◽  
Vol 29 (1) ◽  
pp. 63-68
Author(s):  
Artur Korniłowicz ◽  
Dariusz Surowik
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Prime Numbers ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Composite Number

Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p 2 + 1 = q 2 + r 2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22 n + k (n = 1, 2, . . . ) are composite.

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