Explore the deep structure of mathematics from prime numbers to advancing numbers

In the vast universe of mathematics, there are some fundamental building blocks that not only define the foundations of mathematics, but also lead us to explore unknown territories. This article will take you through an in-depth understanding of the concepts of primes, rings and fields, infinite sequences and limits, and the concepts of decives, reveal the connections between them, and show how Helmut Haser popularized these concepts in the mid-20th century and their importance.

Prime numbers: The cornerstone of natural numbers

A prime number, as a natural number greater than 1, is only divisible by 1 and itself. Like atoms in nature, they are the basic units that make up all the more complex number structures. The fundamental theorem of arithmetic tells us that any natural number greater than 1 can be uniquely decomposed into the product of prime numbers, which demonstrates the centrality of prime numbers in mathematical systems.

Imagine that the essence of a number, no matter how large, can be traced back to a series of prime numbers. The uniqueness of this decomposition makes prime numbers "atoms" in mathematics.

The distribution of prime numbers is both regular and random, and the simplest rule is that all prime numbers are odd except 2. However, there is no fixed pattern for the distances between prime numbers (called prime intervals), which constitutes one of the famous "prime conjectures". This mysterious nature of prime numbers has attracted countless mathematicians and enthusiasts to study, and the discovery of new large prime numbers remains a challenge in the mathematical community to this day.

Rings and Domains: A framework for mathematical structures

Rings and domains give us a high-level way to understand and manipulate numbers. A ring contains a set of elements and the addition and multiplication operations defined on those elements, satisfying a set of basic rules of operation. Domains, on the other hand, add more structure to the ring, so that each non-zero element has an inverse element, allowing us to divide the operation. For example, all rational numbers form a field because every non-zero rational number has an inverse element, which is its reciprocal.

Infinite Sequences and Limits: Quests for infinity

An infinite sequence is a sequence of an infinite number of elements, while a limit describes the behavior when an item in a sequence approaches a certain value. These concepts are the foundation of analytics and help us understand the behavior of functions at a point close to a point or at infinity. For example, with the limit, we can define a derivative (which represents the instantaneous rate of change of a function at a certain point) and an integral (which represents the sum of the areas under the curve of the function).

Advances: Bridges across the number system

Advancing number systems, especially p-advancing numbers, provide mathematicians with a whole new perspective on the properties of numbers. In this system, the size of numbers is not measured in terms of near-infinity or infinitesimal in our usual sense, but is defined in terms of their relationship to some fixed prime number p. This allows mathematicians to use techniques and intuition similar to real number analysis when solving problems related to modular p operations.

Helmut Hasser: a bridge between the past and the future

In the mid-20th century, Helmut Hasse not only deepened our understanding of known mathematical structures, but also opened up avenues for exploring new areas of mathematics by generalizing advance numbers from ordinary primes to more general number systems. His work demonstrates the deep connections between different fields of mathematics and how seemingly unrelated branches of mathematics can be connected through abstract concepts.

The world of mathematics.