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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 ...
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 ...
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 ...
The third topology, introduced by A.M. Kirch, [3] is an example of a countably infinite Hausdorff space that is both connected and locally connected. These topologies also have interesting separation and homogeneity properties. The notion of an arithmetic progression topology can be generalized to arbitrary Dedekind domains.
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.
If collinear points A, B, C, and D are such that D is the harmonic conjugate of C with respect to A and B, then the distances from any one of these points to the three remaining points form harmonic progression. [2] [3] Specifically, each of the sequences AC, AB, AD; BC, BA, BD; CA, CD, CB; and DA, DC, DB are harmonic progressions, where each ...
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} .