Numbers (2)

In the early 17th century, mathematician Pierre de Fermat made several contributions to the field of number theory, including his little theorem which stated that if a is not divisible by a prime p, then a^(p-1) ≡ 1 (mod p). He also noted that if a and b are coprime, then a^2 + b^2 is not divisible by any prime congruent to -1 modulo 4, and that every prime congruent to 1 modulo 4 can be written in the form a^2 + b^2. In addition, Fermat claimed to have shown that there are no solutions to x^n + y^n = z^n for n >= 3, a claim known as Fermat's Last Theorem. Later, in the 18th century, mathematician Leonhard Euler provided proofs for some of Fermat's statements and made his own contributions to the field, including his own generalization of Fermat's little theorem to non-prime moduli and his work on the zeta function. Euler also did work on prime factorization, the distribution of prime numbers, and continued fractions.

The field of number theory was greatly influenced by the works of classical Greek mathematicians such as Plato and Euclid, as well as Indian mathematicians like Āryabhaṭa and Brahmagupta. Later, during the Renaissance, the works of Greek mathematician Diophantus were rediscovered and studied, leading to the development of what is now known as Diophantine equations. In the early 17th century, Pierre de Fermat made significant contributions to the field, including his little theorem and work on integer divisors and perfect numbers. The interest in number theory was renewed in the 18th century with the works of Leonhard Euler, who provided proofs for Fermat's theorems and made advances in the study of quadratic forms. Joseph-Louis Lagrange and Adrien-Marie Legendre also made important contributions, including the law of quadratic reciprocity and the prime number theorem. Finally, Carl Friedrich Gauss' Disquisitiones Arithmeticae solidified the foundations of number theory and introduced new concepts such as congruences.

In the early 1800s, number theory began to be recognized as its own field of study, separate from other areas of mathematics. Along with this development, several subfields of number theory emerged, including analytic and algebraic number theory. As these subfields developed, they drew upon the concepts and techniques of other areas of mathematics, such as complex analysis, group theory, and Galois theory. These subfields also contributed to the development of abstract algebra, ideal theory, and valuation theory.

One of the key turning points in the history of number theory was the work of Dirichlet, who proved a theorem on arithmetic progressions in 1837. This proof introduced the concept of L-functions and made use of asymptotic analysis and limiting processes on real variables. Euler's work in the 1730s, which involved the use of formal power series and non-rigorous limiting arguments, can also be seen as an early use of analytic ideas in number theory. The use of complex analysis in number theory came later, with the work of Riemann on the zeta function in 1859. However, Jacobi's four-square theorem, which was published in 1839 and deals with modular forms, also played a significant role in the development of analytic number theory.

Many questions in both analytic and algebraic number theory remain open and are the focus of ongoing research.

Elementary number theory is a branch of mathematics that studies the properties of integers and integer-valued functions. This includes studying topics such as prime numbers, modular arithmetic, and the distribution of integers. One of the most famous results in elementary number theory is the prime number theorem, which was first proven using complex analysis in 1896. However, an elementary proof was not found until 1949 by Erdős and Selberg. While the results of elementary number theory can often be stated in a way that is understandable to the general public, the proofs of these results can be quite difficult and use a wide range of mathematical tools.

Analytic number theory is a branch of mathematics that investigates the integers using techniques from real and complex analysis. It is characterized by its focus on estimates of size and density rather than identities. Some of the problems studied in analytic number theory include the prime number theorem, the Goldbach conjecture, the Waring problem, and the Riemann hypothesis. The circle method, sieve methods, and L-functions are some of the important tools used in the field. The study of modular forms and automorphic forms is also becoming increasingly important in analytic number theory. It is possible to ask analytic questions about algebraic numbers and use analytic methods to answer them, leading to overlap between algebraic and analytic number theory. An example of this is the study of prime ideals in the field of algebraic numbers, which can be examined using Dedekind zeta functions, a generalization of the Riemann zeta function. In general, analytic number theory involves deriving information about the distribution of a sequence by analyzing the behavior of a complex-valued function.

In the early 19th century, the concepts of ideal numbers and the theory of ideals were developed, alongside valuation theory, as ways to deal with the lack of unique factorization in algebraic number fields. These developments laid the foundation for modern algebraic number theory, which is concerned with the study of algebraic number fields. The classification of possible extensions of a given number field is a difficult and ongoing problem, with the well-understood abelian extensions being classified through the program of class field theory. Current research in the field includes Iwasawa theory and the Langlands program, which aims to generalize class field theory to non-abelian extensions of number fields.

Diophantine geometry is a branch of mathematics that studies the solutions to Diophantine equations, which are equations that seek integer or rational solutions. These equations can be thought of as geometric objects, such as curves or surfaces, in n-dimensional space. Diophantine geometry asks whether these objects have any rational or integer points and, if so, how many and how they are distributed. One important question in this field is whether there are finitely or infinitely many rational or integer points on a given curve or surface. The finiteness or not of these points depends on the genus of the curve, which is determined by the number of holes in the surface defined by the equation when the variables are allowed to be complex numbers. Diophantine geometry is also connected to the field of Diophantine approximations, which involves finding how well a number can be approximated by rationals. This is especially relevant when the number in question is an algebraic number. The study of Diophantine approximations also plays a role in determining whether a number is transcendental, which means it cannot be expressed as the root of a polynomial equation with rational coefficients. Diophantine geometry should not be confused with the geometry of numbers, which is a branch of mathematics that deals with the geometric properties of lattices.