number theory
Number theory is the branch of pure mathematics that studies the properties of numbers in general and integers in particular, as well as various problems derived from their study.
Summary
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- 1 Arithmetic
- 2 Number theory
- 3 Fields
- 4 Elementary number theory
- 5 Sources
- 6 Related link
Arithmetic
Arithmetic is the branch of mathematics that studies certain operations of numbers and their elementary properties. It comes from the Greek origin arithmos and techne, which respectively mean numbers and skill.
number theory
It contains a considerable number of problems that could be understood by “non-mathematicians.” More generally, this field studies the problems that arise with the study of integers. As Jürgen Neukirch quotes : Number theory occupies among the mathematical disciplines an idealized position analogous to that occupied by mathematics itself among the other sciences. The term arithmetic was also used to refer to number theory. This is a fairly old term, although not as popular as in the past. Hence number theory is usually called high arithmetic, although the term has also fallen into disuse. This sense of the term arithmetic should not be confused with elementary arithmetic, or with the branch of logic that studies Peano arithmetic as a formal system. Mathematicians who study number theory are called number theorists.
Fields
Depending on the methods used and the questions to be answered, number theory is subdivided into various branches.
Elementary number theory
In elementary number theory, integers are studied without using techniques from other fields of mathematics. The questions of divisibility, Euclid ‘s algorithm to calculate the greatest common divisor, the factorization of integers as products of prime numbers , the search for perfect numbers and congruences belong to elementary number theory. Typical statements are Fermat’s little theorem and Euler’s theorem that extends it, the Chinese remainder theorem and the law of quadratic reciprocity . In this branch, the properties of multiplicative functions such as the Möbius function and the Euler φ function are investigated ; as well as sequences of integers such as factorials and Fibonacci numbers .
Various questions within elementary number theory seem simple, but require very deep considerations and new approaches, including the following:
- Goldbach’s conjecture that all even numbers are the sum of two prime numbers.
- Twin prime number conjecture about the infinity of the so-called twin prime numbers.
Fermat’s last theorem (proved in 1995 ) Riemann hypothesis on the distribution of the zeros of the Riemann zeta function , closely connected with the problem of the distribution of prime numbers.