Elementary Number Theory

This is  the demo version of my new number theory problem set which contains 307 problems from 2015 - 2016 mathematical competitions and olympiads around the world. If you want the original version, you can download it for a finite price here: https://parvardi.com/downloads/NT2016/

This study is realized to make individuals understand that irrational numbers are expressed by “infinite decimal numbers that do not repeat”. For this, a visual model proposal for the teaching of irrational numbers is presented. This visual model is based on e=2.71828182845904... and π=3,141592653589793238… numbers which are often seen by studenst and teachers in mathematics education and science. The digits of these numbers after the comma cannot be written using repeating or a finite number of digits (infinite non-repeating decimal representation). This is one of the two expressions used to express irrationality in the mathematical society. By this way, based on the number of e and π, a visual model was presented to make the students understand the irrational numbers. To make this visualization, the numbers of first 2500 integer part and decimal digits of e and  were placed on the size of 50 x 50 table in Microsoft Excel and these visualizations were presented on a fountain which was designed by the researcher. Then, all the numbers from 0 to 9 were assigned different colors and each cell of table was colored with the corresponding assigned colors instead of numbers. The colored Excel table made in this way was converted into image format. Then, it was intended to draw attention to these numbers via planning the illumination on the designed fountain and reflecting these colors in the sky or on the wall / floor in the evening.

Bernoulli numbers which are ubiquitous in mathematics, typically appearing either as the Taylor coefficients of x/ tan x or else being very closed to this, as special values of the Riemann zeta function. But they also sometimes appear in other guises and in other combinations. Here we will look at the functional equation satisfied by the Bernoulli numbers. We begin with the less familiar and divergent ordinary generating series β(x) = ∞ n=0

This is the ultimate proof that the Prime Numbers are not Random Numbers as famous Mathematicians believe and claim publicly through presentations you can find on U Tube
Any pupil around the world and a wide public will understand the first 2 pages of the paper below.  Feel free to ask questions by emailing me constantine.adraktas@mit-partners.eu

Arithmetic functions have rather rich properties and very useful tool for both in number theory and algebra. Furthermore, they have many implementations in science such as computer science, cryptography, graph theory etc. The essential arithmetic functions are multiplicative and additive functions. Simply the completely additive functions have ψ(mn)=ψ(m)+ψ(n) and completely multiplicative functions have ψ(mn)=ψ(m)ψ(n), for all natural numbers m and n. On the other hand, ψ is additive if ψ(mn)=ψ(m)+ψ(n) for all coprime natural numbers m and n, or else ψ is multiplicative if ψ(mn)=ψ(m)ψ(n) for all coprime natural numbers m and n. Some fundamental multiplicative functions are sigma function (σ), tau function (τ), Euler totient function, Möbius function, Dedekind psi function etc. In the present study, we investigate interesting properties of these functions by an elementary approaches and we give some applications. We mainly focus on well-known Euler’s totient function and similar arithmetic functions in terms of structure of residue class modulo m. By the help of divisor function (d) and Euler totient function, we obtain an inequality. Then, we derive new inequalities by former ones. Moreover, we use these functions in order to prove some theorems in field theory. Finally, we give some examples and procedures about our results.

This paper contains a new proof of Euler’s theorem, that the only non-trivial integral solution, (α, β), of α2 = β3 +1 is (±3, 2). This proof employs only the properties of the ring, Z, of integers without recourse to elliptic curves and is independent of the methods of algebraic number fields. The advantage of our proof, over Euler’s isolated and other known proofs of this result, is that it charts a common path to a novel approach to the solution of Catalan’s conjecture and indeed of any Diophantine equation. Mathematics Subject Classification: 11R04

The Goldbach Conjecture remains one of the several unsolved mathematical problems today, along with the Twin Prime Conjecture. It is said to be one of the simplest mathematical problems to state yet the most difficult to prove.  In this paper, the author explores a few even numbers about the problem, and, using them as his sample size, juxtaposes his ideas with those of others through literature review. The author explores a solution to the Goldbach conjecture which was put forward about 275 years ago, by a German mathematician, Christian Goldbach, who was a contemporary of the genius German mathematician, Leonhard Euler. Euler in a letter to Goldbach, had confessed at the time that unfortunately, he (Euler) could not prove Goldbach’s mathematical poser. Through fundamental analysis and critical thinking, this author attempts to share his thoughts on how to resolve this long-standing mathematical debacle. The methodology used is basic, experimental, and exploratory, with the use of inferences through recognition of number patterns. We hope that this paper will make a small contribution to the discourse on Goldbach’s conjecture, whose solution has eluded mathematicians for about 275 years.

This paper brings numbers in such a way that both sides of the expressions are with same digits. One side is numbers with power, while other side just with numbers, such as, a^b+c^d +.. =ab +cd+..., etc. The the expressions studies are with positive and negative signs. Work is done for two to five terms expressions. Due to high quantity of numbers, in some cases, powers and bases both are considered bigger than one.

The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact  this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (S sequences) but also on “Smarandache-Coman sequences of primes” (SC sequences), defined by the author as “all sequences of primes obtained from the terms of Smarandache sequences using any arithmetical operation”: the SC sequences presented in this book are related, of course, to concatenation, but in three different ways: the S sequence is obtained by the method of concatenation but the operation applied on its terms is some other arithmetical operation; the S sequence is not obtained by the method of concatenation but the operation applied on its terms is concatenation, or both S sequence and SC sequence are using the method of concatenation. Part Two of this book, “Sequences of primes obtained by the method of concatenation”, brings together 51 articles which aim, using the mentioned method, to highlight sequences of numbers that are rich in primes or are liable to lead to large primes.  The method of concatenation is applied to different classes of numbers, e.g. squares of primes, Poulet numbers, triangular numbers, reversible primes, twin primes, repdigits, factorials, primorials, in order to obtain sequences, possible infinite, of primes. Part Two of this book also contains a paper which lists a number of 33 sequences of primes obtained by the method of concatenation, sequences presented and analyzed in more detail in my previous papers, gathered together in five books of collected papers: “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Two hundred and thirteen conjectures on primes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, “Sequences of integers, conjectures and new arithmetical tools”, “Formulas and polynomials which generate primes and Fermat pseudoprimes”.

People these days know the Universe as a Whole, because not knowing the edge between This and That. Its secret is the secret of a forgotten body, the seventh in the series of multifaceted as Life, Seven. We know six of it now. But the Ancestors knew all seven. // Люди ныне не знают Вселенной как Целого, не зная грани меж Этим и Тем. Тайна  ее есть тайна забытого тела, седьмого в строю многогранных как Жизнь, Семь. Мы знаем шесть ныне. А Пращуры знали все семь.

This is a book on Olympiad Number Theory. It takes a very conceptual approach on the theory and is filled with challenging solved examples and problems with hints.

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List of typos:
https://drive.google.com/file/d/1xBKFZ7JV4PkYLHCFjZKdXVia0zsd2I_3/view?usp=sharing

A short introduction to some connections between music and math.  Polyrhythms may be analyzed using basic number theory concepts such as GCD and LCM.  Tuning using only rational multiples of a fundamental frequency (just intonation) leads to ideas in abstract algebra, including abelian groups, lattices, and factor groups.

Đặc trưng của phương trình kiểu Pell và một số phương trình quy về phương trình kiểu Pell là có tập nghiệm biểu diễn được thông qua các dãy số. Vì vậy từ các phương trình Diophant này ta có thể tạo ra các bài toán dãy số nguyên, cũng như tìm cách giải cho các bài toán này. Ví dụ như trong các đề thi học sinh giỏi Quốc gia môn toán năm 1999 và năm 2012 có hai bài toán về dãy số, hai bài toán này gắn với 2 phương trình Diophant là $u^2-5v^2=-4$ và $x^2-4xy+y^2+2=0$ mà có thể quy về phương trình kiểu Pell $u^2-3v^2=-2$. Khi giải các bài toán này có hai hướng là sử dụng công thức nghiệm của phương trình kiểu Pell hoặc sử dụng phương pháp \textit{Vieta Jumping} \cite{Vite}. Với cách tiếp cận như vậy bài viết này nêu cách tạo ra một số bài toán về dãy số nguyên từ phương trình Diophant.

Cdo kush bie dakord qe π eshte nje nga numrat me te rendesishem ne matematike, kjo sepse π ka aq shume veti, sa qe kur ky numer shfaqet papritmas ne llogaritje, askush nuk duhet te surprizohet padrejtesisht. Per shembull, teorema e njohur nga Euler(Ejler) .

En el siguiente artículo se pretende dar una base tanto para los estudiantes de olimpiadas como para matemáticos aficionados que disfrutan de una buena lectura sobre la teoría elemental de números. Va desde resultados básicos de divisibilidad a aritmética modular con varios ejercicios resueltos de olimpiadas.

Les nombres premiers sont les nombres speciaux pour les mathématiciens car on désire de comprendre les nombres, et la pièce élémentaire d'un nombre, est un nombre premier et il n'existe pas une formule explicite pour les determiner.

Dans ce projet de fin d'études, notre but est de donner une formule pour la distribution des nombres premiers avec le Théorème des Nombres Premieres et demontrer une relation entre la Fonction Zêta de Riemann et le Théorème des Nombres Premieres, (on va le noté TNP).

Les recherches sur les nombres premiers se poursuivent aujhourd'hui encore. Les mathématiciens travaillent pour déterminer le plus grande premier dans l'espoire d'avoir écrit leur noms dans l'annales. Il y a beaucoup des questions ouverts qui soutient les nouveaux recherches.

“How can all of this be true all at the same time?”
This will be the question you will be asking yourself once you discover the amazing inner world hiding behind numbers, as they reveal palindromes, two types of dual characteristics, visible and invisible patterns, perfect plus/minus as well as odd/even balances, and much more. Learn that prime numbers can be organized in a perfect 24-based, yet decimally-based system and aren’t randomly distributed. Discover a whole new way to see numbers as one unified, and I dare say, Intelligent and Logical system. An entire new Number Theory is hereby introduced as well. Find out further aspects hiding behind the Fibonacci numbers, and similarly found ratios point to the square roots of whole numbers. A bonus chapter reveals the number present in our solar system.

Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is also known as the golden mean, golden section, golden number or divine proportion.It is usually denoted by the Greek letter  (phi).

Numerele naturale au fascinat dintotdeauna omenirea, ce le-a considerat, pe bună dreptate, ca fiind mai mult decât mijloace de a studia cantităţile, le-a considerat entităţi având o personalitate proprie. Mistica tuturor popoarelor abundă de proprietăţi supranaturale atribuite numerelor. Într-adevăr, pare că, pe măsură ce le cercetezi mai adânc, descoperi că au „consistenţă”, că, departe de a fi o creaţie conceptuală a omului, un simplu instrument la îndemâna sa, se conduc de fapt după legităţi proprii, pe care nu-ţi permit să le influenţezi, ci doar să le descoperi. Printre primii oameni care au simţit acest lucru se numără Pitagora, ce a început prin a cerceta numerele şi a sfârşit prin a întemeia o mişcare religioasă puternic fondată pe simbolistica numerelor. Pasiunea sa pentru numerele naturale era atât de mare (accepta, totuşi, şi existenţa numerelor raţionale, ce sunt, în fond, tot un raport de numere naturale), încât circulă o legendă conform căreia şi-ar fi înecat un discipol pentru „vina” de a-i fi relevat existenţa numerelor iraţionale. Mult mai aproape pe scara istoriei, în secolul XIX, matematicianul german Leopold Kronecker este creditat a fi spus: „Dumnezeu a creat numerele naturale; toate celelalte sunt opera omului”. Departe de considerente de filozofie a matematicii, ne-am limitat enciclopedia la numerele întregi pentru că am considerat că este un domeniu îndeajuns de vast în sine pentru a face obiectul unei astfel de lucrări. Am împărţit enciclopedia în două părţi, „Clase de numere” (se subînţelege, întregi), respectiv „Clase de prime şi pseudoprime”, prima cuprinzând principalele clase de numere cu care se operează actualmente în teoria numerelor (ramura matematicii ce studiază în principal numerele întregi), cea de-a doua câteva tipuri consacrate de prime (numere care au „sfidat” dintotdeauna matematicienii prin rezistenţa lor în a se lăsa înţelese şi ordonate) şi tipurile cunoscute de pseudoprime (o categorie aparte de numere,  ce împart însă multe atribute cu numerele prime). Am încercat să folosim noţiuni cât mai simple şi operaţii elementare pentru a nu îndepărta cititorii prin simboluri şi denumiri de funcţii; din acelaşi motiv am definit toate clasele de numere doar în sistemul comun, zecimal, şi nu am considerat clasele de numere care, deşi întregi, se definesc apelând la numere iraţionale sau complexe.