The Prime Glossary: arithmetic sequence

arithmetic sequence
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An arithmetic sequence (or arithmetic progression) is a sequence (finite or infinite list) of real numbers for which each term is the previous term plus a constant (called the common difference). For example, starting with 1 and using a common difference of 4 we get the finite arithmetic sequence: 1, 5, 9, 13, 17, 21; and also the inifinite sequence

1, 5, 9, 13, 17, 21, 25, 29, . . ., 4n+1, . . .

In general, the terms of an arithmetic sequence with the first term a₀ and common difference d, have the form a_n = dn+a₀ (n=0,1,2,...). If a₀ and d are relatively prime positive integers, then the corresponding infinite sequence contains infinitely many primes (see Dirichlet's theorem on primes in arithmetic progressions).

An important example of this is the following two arithmetic sequences:

1, 7, 13, 19, 25, 31, 37, ...
5, 11, 17, 23, 29, 35, 41, ...

Together these two sequences contain all of the primes except 2 and 3.

A related question is how long of a arithmetic sequence can we find all of whose members are prime. Dickson's conjecture says the answer should be arbitrarily long--but finding long sequences of primes is quite difficult. It is fairly easy to heuristically estimate how many such primes sequences there should be for any given length--Hardy and Littlewood first did this in 1922 [HL23]. In 1939, van der Corput showed that there are infinitely many triples of primes in arithmetic progression [Corput1939]. Finally, in 2004, Green and Tao [GT2004a] showed that there are indeed arbitrarily long sequences of primes and that a k-term one occurs before [GT2004b]:

2^{2^{2^{2^{2^{2^{2^{2^100k}}}}}}}

Obviously this is not optimal! It is conjectured that it actually occurs before k!+1 [Kra2005].

The longest known arithmetic sequence of primes is currently of length 25, starting with the prime 6171054912832631 and continuing with common difference 366384*23#*n, found by Chermoni Raanan and Jaroslaw Wroblewski in May 2008.

The longest known sequence of consecutive primes in arithmetic progression is ten starting with the 93-digit prime

100 9969724697 1424763778 6655587969 8403295093 2468919004 1803603417 7589043417 0334888215 9067229719,

and continuing with common difference 210. (See Tony Forbes' web page for more information.)

In August 2000 David Broadhurst found the smallest arithmetic progression of titanic primes of length three:

10⁹⁹⁹+61971, 10⁹⁹⁹+91737, 10⁹⁹⁹+121503;

and of length four:

10⁹⁹⁹+2059323, 10⁹⁹⁹+2139213, 10⁹⁹⁹+2219103, 10⁹⁹⁹+2298993.

See Also: GeometricSequence

Related pages (outside of this work)

Jens Kruse Andersen's excellent Primes in Arithmetic Progression Records
Ten consecutive primes in arithmetic progression by Tony Forbes

References:

Chowla44
S. Chowla, "There exists an infinity of 3--combinations of primes in A. P.," Proc. Lahore Phil. Soc., 6 (1944) 15--16. MR 7,243l
Corput1939
A. G. van der Corput, "Über Summen von Primzahlen und Primzahlquadraten," Math. Ann., 116 (1939) 1--50.
DFLMNZ1998
H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson and P. Zimmermann, "Ten consecutive primes in arithmetic progression," Math. Comp., 71:239 (2002) 1323--1328 (electronic). MR 1 898 760 (Abstract available)
DN97
H. Dubner and H. Nelson, "Seven consecutive primes in arithmetic progression," Math. Comp., 66 (1997) 1743--1749. MR 98a:11122 (Abstract available)
GT2004a
B. Green and T. Tao, "The primes contain arbitrarily long arithmetic progressions," Ann. Math., preprint. Available from http://arxiv.org/abs/math/0404188v6.
GT2004b
B. Green and T. Tao, "A bound for progressions of length k in the primes," (2004) preprint.
Guy94 (A6)
R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, NY, ISBN 0-387-94289-0. 1994. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references.]
HL23
G. H. Hardy and J. E. Littlewood, "Some problems of `partitio numerorum' : III: on the expression of a number as a sum of primes," Acta Math., 44 (1923) 1-70. Reprinted in "Collected Papers of G. H. Hardy," Vol. I, pp. 561-630, Clarendon Press, Oxford, 1966.
Kra2005
B. Kra, "The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view," Bull. Amer. Math. Soc., (2005) Posted October 6, 2005. PII: S 0273-0979(05)01086-4 (to appear in print). (Abstract available)
LP1967a
L. J. Lander and T. R. Parkin, "Consecutive primes in arithmetic progression," Math. Comp., 21 (1967) 489.
Rose94
H. E. Rose, A course in number theory, second edition, Clarendon Press, New York, pp. xvi+398, ISBN 0-19-853479-5; 0-19-852376-9. 1994. MR 96g:11001 (Annotation available)