Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In 6th century BCE, ancient Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can b ...
The first statement in terms of logarithmically convex functions Khabibullin's conjecture (version 1, 1992). Let \displaystyle S be a non-negative increasing function on the half-line Relation to Eule ...
The irrationality measure (or irrationality exponent or approximation exponent or Liouville–Roth constant) of a real number x is a measure of how "closely" it can be approximated by rationals. Genera ...
The name "transcendental" comes from Leibniz in his 1682 paper where he proved that sin(x) is not an algebraic function of x. Euler was probably the first person to define transcendental numbers in th ...
The rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because x=a/b is the root of bx-a.The quadratic surds (irrational roots of a ...
The proof presented here was arranged by Ryll-Nardzewski (1951) and is much simpler than Khinchin's original proof which did not use ergodic theory.Since the first coefficient a0 of the continued frac ...
Catalan's constant occurs frequently in relation to the Clausen function, the Inverse tangent integral, the Inverse sine integral, Barnes G-function, as well as integrals and series summable in terms ...
The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Enestr?m Index 43). Euler used the notations C and O for the ...
Pompeiu, Dimitrie (1929), "Sur certains systèmes d'équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables", Comptes Rendus de l'Académie des Sciences. Série I. ...
Let P(x)\in\mathbb{Z} be an irreducible monic polynomial of degree D.Smyth proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying x^DP(x ...
The conjecture is as follows:Given any n complex numbers z1,...,zn which are linearly independent over the rational numbers Q, the extension field Q(z1,...,zn, exp(z1),...,exp(zn)) has transcendence d ...
Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implic ...
Integral basis An integral basis for a number field F of degree n is a setB = {b1, …, bn}of n algebraic integers in F such that every element of the ring of integers OF of F can be written uniquely a ...
Let K/k be an abelian extension of global fields, and let S be a set of places of k containing the Archimedean places and the prime ideals that ramify in K/k. The S-imprimitive equivariant Artin L-fun ...
Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero x of R can be written as a product (an empty product if x is a unit) of irreducible elements pi of ...
