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For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a 1 {\displaystyle a_{1}} and the common difference of successive members is d {\displaystyle d} , then the n {\displaystyle n} -th term of the sequence ( a n {\displaystyle a_{n ...
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
Dirichlet, P. G. L. (1837), "Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält" [Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without ...
There has been separate computational work to find large arithmetic progressions in the primes. The Green–Tao paper states 'At the time of writing the longest known arithmetic progression of primes is of length 23, and was found in 2004 by Markus Frind, Paul Underwood, and Paul Jobling: 56211383760397 + 44546738095860 · k ; k = 0, 1 ...
Recamán's sequence: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ... "subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence. A005132: Look-and ...
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.
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions , vectors , matrices , polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Catalan discovered this sequence after the discovery of the primality of = by Lucas in 1876. [ 1 ] [ 5 ] Catalan conjectured that they are prime "up to a certain limit". Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge.
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