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An arithmetic progression or arithmetic sequence is a ... The sum of the members of a finite arithmetic progression is called an arithmetic series. For example ...
Dirichlet's theorem on arithmetic progressions. 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 ...
In number theory, primes in arithmetic progressionare any sequenceof at least three prime numbersthat are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by an=3+4n{\displaystyle a_{n}=3+4n}for 0≤n≤2{\displaystyle 0\leq n\leq 2}. According to the Green–Tao theorem, there ...
v. t. e. In mathematics, an arithmetico-geometric sequence is the result of element-by-element multiplication of the elements of a geometric progression with the corresponding elements of an arithmetic progression. The n th element of an arithmetico-geometric sequence is the product of the n th element of an arithmetic sequence and the n th ...
Green–Tao theorem. In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi ...
This is stronger than Dirichlet's theorem on arithmetic progressions (which only states that there is an infinity of primes in each class) and can be proved using similar methods used by Newman for his proof of the prime number theorem. [32] The Siegel–Walfisz theorem gives a good estimate for the distribution of primes in residue classes.
Faulhaber's formula concerns expressing the sum of the p -th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n. The first few examples are well known. For p = 0, we have For p = 1, we have the triangular numbers For p = 2, we have the square pyramidal numbers. The coefficients of Faulhaber's formula in its ...
An arithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference. [83] This difference is called the modulus of the progression. [84] For example, 3, 12, 21, 30, 39, ..., is an infinite arithmetic progression with modulus 9.
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