Search results
Results from the WOW.Com Content Network
Proof without words of the arithmetic progression formulas using a rotated copy of the blocks. An arithmetic progression or arithmetic sequence 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 ...
Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have also been called non-averaging sets, [ 1 ] [ 2 ] but this term has also been used to denote a set of integers none of which can be obtained as the average of any subset of the other numbers. [ 3 ]
P(n) = P(n − 2) + P(n − 3) for n ≥ 3, with P(0) = P(1) = P(2) = 1. A000931: Euclid–Mullin sequence: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ... a(1) = 2; a(n + 1) is smallest prime factor of a(1) a(2) ⋯ a(n) + 1. A000945: Lucky numbers: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, ... A natural number in a set that is filtered by ...
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} .
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 ...
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 ...
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 ...
The problem of obtaining bounds in the k=3 case of Szemerédi's theorem in the vector space is known as the cap set problem. The Green–Tao theorem asserts the prime numbers contain arbitrarily long arithmetic progressions. It is not implied by Szemerédi's theorem because the primes have density 0 in the natural numbers.