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For example, if s=2, then ๐(s) is the well-known series 1 + 1/4 + 1/9 + 1/16 + …, which strangely adds up to exactly ๐²/6. When s is a complex number—one that looks like a+b๐, using ...
Logical reasoning is a form of thinking that is concerned with arriving at a conclusion in a rigorous way. [1] This happens in the form of inferences by transforming the information present in a set of premises to reach a conclusion.
Oppermann's conjecture is an unsolved problem in mathematics on the distribution of prime numbers. [1] It is closely related to but stronger than Legendre's conjecture, Andrica's conjecture, and Brocard's conjecture. It is named after Danish mathematician Ludvig Oppermann, who announced it in an unpublished lecture in March 1877. [2]
The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Semantic completeness is the converse of soundness for formal systems. A formal system is complete with respect to tautologousness or "semantically complete" when all its tautologies are theorems, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system ...
Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers. Determining whether a given initial point with rational coordinates is periodic, or whether it lies in the basin of attraction of a given open set, in a piecewise-linear iterated map in two dimensions ...
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic.More precisely, a cardinal κ is strongly inaccessible if it satisfies the following three conditions: it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and < implies <.
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