We start by discussing the concept of density - fundamental to this article - and we provide a rigorous definition and possibly a new result , for measuring the density of an infinite subset of integers. Then we investigate forward difference sequences to address the issue arising from zero singular densities: in the case of prime numbers, it leads to beautiful, mysterious conjectures.
Finally, we provide an algorithm that computes quantities related to densities, for a number of integer families, including prime numbers, and integers that are sum of two squares. The last section discusses potential areas for additional research, such as a probabilistic number theory , generating functions for composite numbers possibly leading to a generating function for primes as well as strong abnormalities in the continued fraction expansions for many constants, including for the special mathematical constants Pi, K, e, and other transcendental numbers mentioned in this article.
Likewise, because the prime numbers less or equal to n are becoming increasingly rare as n grows, the density of this set the prime numbers is asymptotically equal to 0. Here "log" stands for the natural logarithm of base e. Other examples of interesting numbers are the sum of 1, 2, 3, or 4 square numbers.
Square numbers are so rare among integers, far more rare than primes, that their density is 0. Numbers that are the sum of four squares are very abundant.
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Actually any integer can be written as the sum of four square integers, usually in several different ways. Thus the density of this set integers that are sum of four squares is exactly 1. What about integers that are the sum of two squares? They are more abundant than prime numbers, yet rare enough that their density is still 0. This provides another way to estimate Pi. Interestingly, the same algorithm can be used to get all the composite numbers, and thus all the prime numbers. So in some sense, both prime numbers and numbers that are sum of two squares belong to a same family of integers - each one being a sub-type of the general family.
And if we consider numbers that are the sum of two primes, rather than the sum of two squares, what density should we expect to get? Remember, primes are far more numerous than square numbers.
MAT315H1: Introduction to Number Theory
Sums of two squares is a particular case of Waring's problem. This is a topic with numerous well-known conjectures. For a bunch of conjectures about prime numbers, click here. In many cases, we would like to compare two densities, even when they are both equal to zero: for instance, integers that are sum of two squares, versus primes. For n large enough , let's define. For an infinite subset S of positive integers, let's define z n ; S as the number of elements in S that are less than, or equal to n.
Here we conjecture that for any such subset S, there exist coefficients p , q , r , s , That is, the density of the set S is characterized by the coefficients p , q , r , s , Moreover, the family of limiting functions n , log n , log log n , log log log n etc. This statement is still a conjecture, but it is true for all the useful sets that I have ever seen, including all the examples in this article. The bivariate vector U, V uniquely defines the density of the set S in question, and can be used to compare densities. Let's D n be the set of numbers that are divisible by 2 or 3 or 5 or What is the density of D n?
Department of Mathematics - Math - Introduction to Number Theory
If we denote as d n the density of D n , how fast does d n converge to 1 as n tends to infinity? Can you find a recurrence formula for d n? It would not work if instead of these numbers, we had chosen say 2, 3, 5, 7, and Using the Bonferroni inequalities , one can find nested lower and upper bounds for 1 - d n.
In this section, two more conjectures about prime numbers will be discussed. A new result presented here is that one of these conjectures is not specific to prime numbers, and can easily be generalized. The purpose of this section is to obtain a meaningful non-singular density measurement when the original set be it the prime numbers, or numbers that are sum of two squares has a zero that is, singular density. In order to achieve this goal, we look at a data set D S derived from the original set S.
In particular, we look at the data set generated by forward differences. For instance, the set S of squares 0, 1, 4, 9, 16, Since the book is quite long, an instructor for a one-semester course would need to choose specific topics from the text to cover. There are a few sections indicated that are not required for future material.
However, even among the remaining sections, an instructor would need to carefully choose sections that include all necessary prerequisite material. Depending on a course's focus, this could be done fairly easily. Each chapter is written in a logical manner, referencing previous material as needed.
The book jumps from chapters on purely algebraic topics to those focused on applications. For this reason, an instructor may want to choose certain sections in a chapter to cover as prerequisites for an application, instead of covering the material linearly. As promised by the title, the book gives a very nice overview of a side range of topics in number theory and algebra primarily the former, but with quite a bit of attention to the latter as well , with special emphasis to the areas in which As promised by the title, the book gives a very nice overview of a side range of topics in number theory and algebra primarily the former, but with quite a bit of attention to the latter as well , with special emphasis to the areas in which computational techniques have proved useful.
There is a very good index and glossary and a good review of notation and basic facts in the first chapter.
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The format of the book makes it especially easy to update as advances in the subjects occur, particularly computational advances. References are given to websites as well as books. The prose is very lucid and easy to follow. Many examples are given and difficult ideas are introduced gradually. The many relationships between number theory and algebra are explored in detail, each subject yielding important insights into and applications of the other.
No jargon is used and terminology is carefully explained. The book has a very consistent framework and a nice flow from one chapter to the next. As mentioned above, relationships between the two subjects of the title are emphasized. The book is nicely broken up into manageable sections that would fit well into a lecture course. Interdependences among chapters are clearly indicated.
The topics are presented clearly and logically with relationships among them clearly pointed out and discussed in detail. All pages display very well on my screen, with no legibility or distortion issues that I could see. This is not relevant for a mathematics text, but I saw nothing that would be offensive to a reader of any ethnic background. The text is so comprehensive that it feels overwhelming. The author wanted to include all of the mathematics required beyond a standard calculus sequence. However, the mathematical maturity required to read and learn from this text is quite However, the mathematical maturity required to read and learn from this text is quite high.
The first two chapters cover much of a standard undergraduate course in number theory, built up from scratch.
However, it almost completely lacks numerical examples and computational practice for the students, which would give those new to the material time and experience in which to digest, assimilate, and understand the material. I would think that a book targeted at this level of mathematical sophistication would assume students are comfortable with for example the most basic notions of group theory or the idea of equivalence classes. I can't imagine an appropriate audience for this text: one with the ability to read and work entirely at this abstract level but without any or most of the mathematical preparation provided in at least half the chapters.
I found no mathematical errors. These awards serve to highlight outstanding achievements, contributions, and service to the Earth and Space science community.
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