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Arithmetic progression. An arithmetic progression or arithmetic sequence (AP) is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression. For instance, the sequence 5, 7, 9, 11, 13 ...
For integer k ≥ 3, an AP-k (also called PAP-k) is any sequence of k primes in arithmetic progression. An AP- k can be written as k primes of the form a · n + b, for fixed integers a (called the common difference) and b, and k consecutive integer values of n. An AP- k is usually expressed with n = 0 to k − 1.
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d.
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.
Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory.Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color.
t. e. In mathematics, arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put plainly, the n th term of an arithmetico-geometric sequence is the product of the n th term of an arithmetic sequence and the n th term of a geometric one. [1 ...
Roth's theorem on arithmetic progressions (infinite version): A subset of the natural numbers with positive upper density contains a 3-term arithmetic progression. An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of [ N ] = { 1 , … , N } {\displaystyle [N ...
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. [ 3][ 4][ 5] For example, The theorem says two things about this example: first ...