Combinatorial and Additive Number Theory III

Combinatorial and additive number theory are two closely related branches of mathematics that study the properties of integers. Combinatorial number theory is concerned with the counting and enumeration of integer sequences, while additive number theory is concerned with the properties of sums and differences of integers.

This article will provide an overview of some of the main topics in combinatorial and additive number theory. We will begin with a discussion of the basics of combinatorial number theory, including the use of generating functions and recurrence relations. We will then discuss some of the main topics in additive number theory, including the Goldbach conjecture, Waring's problem, and the Erdős–Straus conjecture.

Combinatorial and Additive Number Theory III: CANT, New York, USA, 2024 and 2024 (Springer Proceedings in Mathematics & Statistics Book 297)

by Melvyn B. Nathanson

4.2 out of 5

Language	:	English
File size	:	5304 KB
Screen Reader	:	Supported
Print length	:	241 pages
X-Ray for textbooks	:	Enabled

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Combinatorial Number Theory

Combinatorial number theory is concerned with the counting and enumeration of integer sequences. One of the most important tools in combinatorial number theory is the generating function. A generating function is a formal power series that encodes the number of solutions to a particular combinatorial problem.

Generating functions have a wide range of applications in additive and combinatorial number theory since most number-theoretic functions have multiplicative properties, which in turn are reflected in their generating functions. For example, consider the generating function for integers with exactly two prime factors:

$$f(x) = \prod_{p \text{ prime}}\frac{1}{1 - px^2}.$$

This generating function can be used to count the number of integers with exactly two prime factors up to a given bound. The coefficient of $x^n$ in $f(x)$ is equal to the number of integers with exactly two prime factors up to $n$.

Another important tool in combinatorial number theory is the recurrence relation. A recurrence relation is an equation that expresses the value of a sequence in terms of its previous values. Recurrence relations can be used to solve a wide range of combinatorial problems.

For example, consider the sequence of Fibonacci numbers, which is defined by the recurrence relation

$$F_n = F_{n-1}+ F_{n-2},$$

with initial conditions $F_0 = 0$ and $F_1 = 1$. The Fibonacci numbers have a wide range of applications in mathematics and computer science.

Additive Number Theory

Additive number theory is concerned with the properties of sums and differences of integers. One of the most famous problems in additive number theory is the Goldbach conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes. The Goldbach conjecture has been unsolved for over 250 years, and it is one of the most famous unsolved problems in mathematics.

Another important problem in additive number theory is Waring's problem, which asks for the minimum number of summands needed to represent any integer as a sum of $k$th powers. For example, Waring's problem asks for the minimum number of cubes needed to represent any integer as a sum of cubes. Waring's problem has been solved for $k = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 42, 48, 60, 72, 84, 90, 108, 120, 168, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 15120, 45360, 136080, 272160, 544320, 1088640, 4354560, 8709120, 21772800, 65318400, 72576000, 290304000, 1161216000, 3483648000, 17418240000, 87091200000.$$

The Erdős–Straus conjecture is another important problem in additive number theory. The Erdős–Straus conjecture states that for any integer $n$, there exists a constant $c$ such that any set of $n$ integers contains a subset of $c$ integers whose sum is divisible by $n$. The Erdős–Straus conjecture has been unsolved for over 60 years, and it is one of the most famous unsolved problems in additive number theory.

Combinatorial and additive number theory are two closely related branches of mathematics that study the properties of integers. Combinatorial number theory is concerned with the counting and enumeration of integer sequences, while additive number theory is concerned with the properties of sums and differences of integers. In this article, we have provided an overview of some of the main topics in combinatorial and additive number theory.

We have discussed the use of generating functions and recurrence relations in combinatorial number theory. We have also discussed some of the main problems in additive number theory, including the Goldbach conjecture, Waring's problem, and the Erdős–Straus conjecture. These are just a few of the many topics that are studied in combinatorial and additive number theory. These branches of mathematics are rich and active areas of research, and they have a wide range of applications in other areas of mathematics and computer science.

Combinatorial and Additive Number Theory III: CANT, New York, USA, 2024 and 2024 (Springer Proceedings in Mathematics & Statistics Book 297)

by Melvyn B. Nathanson

4.2 out of 5

Language	:	English
File size	:	5304 KB
Screen Reader	:	Supported
Print length	:	241 pages
X-Ray for textbooks	:	Enabled

FREE