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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.
Smale's problems is a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 [1] and republished in 1999. [2] Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century.
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number , or it is not, in which case it is a prime number .
The prime number theorem asserts that an integer m selected at random has roughly a 1 / ln m chance of being prime. Thus if n is a large even integer and m is a number between 3 and n / 2 , then one might expect the probability of m and n − m simultaneously being prime to be 1 / ln m ln(n − m) .
Problem #6 at IMO 1988: Let a and b be positive integers such that ab + 1 divides a 2 + b 2. Prove that a 2 + b 2 / ab + 1 is a perfect square. [8] [9] Fix some value k that is a non-square positive integer. Assume there exist positive integers (a, b) for which k = a 2 + b 2 / ab + 1 .
However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation + =, or, what is the same, for finding the quantities whose existence is assured by the Chinese remainder theorem) it first appears in the works of Āryabhaṭa (5th–6th century CE) as an algorithm called kuṭṭaka ...
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