Elements of number theory as a mathematical basis for solving problems for processing integer numbers information

The article describes the methodology for preparing schoolchildren to solve problems for processing integer numbers information, where the use of a brute force algorithm is not rational. The statements of number theory necessary to solve the problems of fnding numbers with a given number of divisors of a certain form are formulated; their evidence is given. A set of exercises is presented that promotes the independent "discovery" of the listed statements by schoolchildren and the formation of a skill to identify the structure of a number with given properties. At the end of the article, algorithms and programs are presented to solve problem No. 25, presented in the training control and measuring materials of the Unifed State Exam in informatics in the 2020/2021 academic year. The exercise sets described in the article can be useful to both informatics teachers and schoolchildren preparing for the Unifed State Exam in informatics.

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NOTE ON THE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718000734 ◽

2018 ◽

Vol 99 (1) ◽

pp. 1-9

Author(s):

ADRIAN W. DUDEK ◽

ŁUKASZ PAŃKOWSKI ◽

VICTOR SCHARASCHKIN

Keyword(s):

Number Theory ◽

Asymptotic Formula ◽

Tauberian Theorem ◽

Fixed Integer ◽

Quadratic Polynomials ◽

Number Of Divisors ◽

The Relationship

Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$.

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Asymptotic formulas for certain arithmetic functions

Mathematica Slovaca ◽

10.2478/s12175-008-0075-2 ◽

2008 ◽

Vol 58 (3) ◽

Author(s):

M. Garaev ◽

M. Kühleitner ◽

F. Luca ◽

W. Nowak

Keyword(s):

Number Theory ◽

Positive Integer ◽

Arithmetic Functions ◽

Asymptotic Formulas ◽

Number Of Divisors ◽

Second Moments

AbstractThis is an extended summary of a talk given by the last named author at the Czecho-Slovake Number Theory Conference 2005, held at Malenovice in September 2005. It surveys some recent results concerning asymptotics for a class of arithmetic functions, including, e.g., the second moments of the number-of-divisors function d(n) and of the function r(n) which counts the number of ways to write a positive integer as a sum of two squares. For the proofs, reference is made to original articles by the authors published elsewhere.

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Euler: The Mathematics of Musical Sadness

Music and the Making of Modern Science ◽

10.7551/mitpress/9780262027274.003.0010 ◽

2014 ◽

Author(s):

Peter Pesic

Keyword(s):

Number Theory ◽

Lower Degree ◽

Web Browsers ◽

Great Mathematician ◽

Mathematical Basis ◽

Bridge Problem ◽

Leonhard Euler ◽

Basic Concepts ◽

New Criterion ◽

Numerical Complexity

Throughout his life, the great mathematician Leonhard Euler spent most of his free time on music, to which he devoted his first book. This chapter discusses how he reformulated the ordering of musical intervals on a new mathematical basis. For this purpose, Euler devised a “degree of agreeableness” that numerically indexed musical intervals and chords, replacing ancient canons of numerical simplicity with a new criterion based on pleasure. Euler applied this criterion (and Aristotle’s teachings about the pleasure of tragedy) to argue that minor intervals and chords evoke sadness through their greater numerical complexity, hence lower degree of agreeableness than the major. This work involved extensive attention to ratios and numerical factorization immediately preceding his subsequent interest in continued fractions and number theory. Having devised a new kind of index, Euler was prepared to put forward indices that would address novel problems like the Königsberg bridge problem and the construction of polyhedra, basic concepts of what we now call topology. Throughout the book where various sound examples are referenced, please see http://mitpress.mit.edu/musicandmodernscience (please note that the sound examples should be viewed in Chrome or Safari Web browsers).

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PENINGKATAN HASIL BELAJAR MAHASISWA PADA PEMBELAJARAN TEORI BILANGAN DENGAN PEMBELAJARAN TIPE STAD

Jurnal Pendidikan Matematika Universitas Lampung ◽

10.23960/mtk/v9i4.pp514-530 ◽

2021 ◽

Vol 9 (4) ◽

pp. 514-530

Author(s):

M Coesamin

Keyword(s):

Number Theory ◽

Mathematics Education ◽

Cooperative Learning ◽

Learning Outcomes ◽

Student Learning Outcomes ◽

Test Results ◽

Education Students ◽

Education Department ◽

Study Program ◽

Academic Year

This action research aims to improve the learning outcomes of Number Theory in Mathematics Education students in the Even semester of the 2019/2020 academic year through STAD-type cooperative learning. The research was conducted at the Mathematics Education Study Program, MIPA Education Department, FKIP University of Lampung. Data were collected through tests and also activity observations to see their contribution to the learning outcomes. The test was carried out after three lessons were carried out at the end of each cycle. The final cycle test was conducted to see student learning outcomes after the implementation of STAD-type cooperative learning. Based on the test results, individual improvement points, awards are determined, as well as determine the increase in learning outcomes of participants in each cycle. The results of data analysis concluded that STAD-type cooperative learning can improve Number Theory learning outcomes in Mathematics Education students in the even semester of the 2019/2020 academic year. The increase in learning outcomes is greater than the increase in student activity in lectures.

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