Number Theory Problems

Partially or totally unsolved questions in number theory and geometry especially, such as coloration problems, elementary geometric conjectures, partitions, generalized periods of a number, length of a generalized period, arithmetic and geometric progressions are exposed.

Independently published, 2018

This challenging book contains fundamentals of elementary number theory as well as a huge number of solved problems and exercises. The authors, who are experienced mathematical olympiad teachers, have used numerous solved problems and examples in the process of presenting the theory. Another point which has made this book self-contained is that the authors have explained everything from the very beginning, so that the reader does not need to use other sources for definitions, theorems, or problems. On the other hand, Topics in Number Theory introduces and develops advanced subjects in number theory which may not be found in other similar number theory books; for instance, chapter 5 presents Thue's lemma, Vietta jumping, and lifting the exponent lemma (among other things) which are unique in the sense that no other book covers all such topics in one place. As a result, this book is suitable for both beginners and advanced-level students in olympiad number theory, math teachers, and in general whoever is interested in learning number theory.

The goal of these several lectures is to discuss in more details some properties of integers. In what follows Z = {0, 1, −1, 2, −2, 3, −3,. .. , n, −n,. .. } will denote the set of all integers and it will be our universe of discourse. By N = {1, 2, 3,. .. , n,. . .} we denote the set of positive integers. If otherwise is stated, letters a, b, c,. .. , x, y, z will be used to represent integers only, and we will often allow ourselves not to mention this in the future. We do not give a formal definition of integers, and assume that the reader is well familiar with their basic properties, such as: We also assume that the First and the Second Principles of Mathematical Induction are valid methods of proving statements of the form ∀n ∈ Z ≥n0 [P (n)], where Z ≥n0 is the set of all integers greater or equal to an integer n 0. We will also use The Well-Ordering Axiom for Z ≥n0 : Every non-empty subset of Z ≥n0 contains unique smallest element. It can be shown that the Well-Ordering Axiom is equivalent to each of the two Principles of Mathematical Induction. By |n| we denote the absolute value of n, which is equal to n if n ≥ 0, and is −n if n < 0 (e.g. |5| = 5, |0| = 0, | − 7| = −(−7) = 7). For every two integers a, b, |ab| = |a| • |b|.

1996

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In this paper, we will discuss two questions, the primality test for Fermat number Fn and several fundamental problems in transfinite number theory. For the primality test for Fermat number Fn, this paper introduces the method of primality test for Fermat number Fn and the Fermat factorization theorem. Meanwhile, the factorization theorem is used to write operation procedure, so that we can distinguish whether 2n+2t+1 is the divisor of Fn or not. For some of the fundamental problems in transfinite number theory, this paper introduces the concept of countable expansion, and then inspects countability of the set of real numbers from the aspect of expansion. Besides, we analyze the cardinal number of linear point set, deriving the fact that their cardinal number possesses invariance.

2014

We announce a number of conjectures associated with and arising from a study of primes and irrationals in R. All are supported by numerical verification to the extent possible. This is an unpublished updated version as of August 13, 2014.

In this paper we shall formulate some open problems, conjectures in the field of number theory. Some of them were formulated earlier in one of my papers [1], [2], [3], [4].

2013

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

ISBN ().444.()()()71·2 250 Problems, in Elementary Number Theory .-WACLAW SIERPINSKI "250 Problems in Elementary Number Theory" presents problems and their solutions in five specific areas of this branch of mathematics: divisibility of numbers, relatively prime numbers, arithmetic progressions, prime and composite numbers, and Diophantic equations. There is, in addition, a section of miscellaneous problems. Included are problems on several levels of difficulty-some are relatively easy, others rather complex, and a number so abstruse that they originally were the subject of scientific research and their solutions are of comparatively recent date. All of the solutions are given thoroughly and in detail; they contain information on possible generalizations of the given problem and further indicate unsolved problems associated with the given problem and solution.