Let the number of Primes of the form less than be denoted . Then

where is the Logarithmic Integral and is the Totient Function.

Let be an increasing arithmetic progression of Primes with minimal difference . If a Prime does not divide , then the elements of must assume all residues modulo , specifically, some element of must be divisible by . Whereas contains only primes, this element must be equal to .

If (where is the Primorial of ), then some prime does not divide , and that prime is in . Thus, in order to determine if has , we need only check a finite number of possible (those with and containing prime ) to see if they contain only primes. If not, then . If , then the elements of cannot be made to cover all residues of any prime . The Prime Patterns Conjecture then asserts that there are infinitely many arithmetic progressions of primes with difference .

A computation shows that the smallest possible common difference for a set of or more Primes in arithmetic progression
for , 2, 3, ... is 0, 1, 2, 6, 6, 30, 150, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 510510, ... (Sloane's A033188, Ribenboim
1989, Dubner and Nelson 1997, Wilson). The values up to are rigorous, while the remainder are lower bounds which
assume the validity of the Prime Patterns Conjecture and are simply given by , where is the th
Prime. The smallest first terms of arithmetic progressions of primes *with minimal differences* are 2, 2, 3, 5, 5,
7, 7, 199, 199, 199, 60858179, 147692845283, 14933623, ... (Sloane's A033189; Wilson).

Smaller first terms are possible for nonminimal -term progressions. Examples include the 8-term progression for , 1, ..., 7, the 12-term progression for , 1, ..., 11 (Golubev 1969, Guy 1994), and the 13-term arithmetic progression for , 1, ..., 12 (Guy 1994).

The largest known set of primes in Arithmetic Sequence is 22,

for , 1, ..., 21 (Pritchard

The largest known sequence of *consecutive* Primes in Arithmetic Progression (i.e., all the numbers between the
first and last term in the progression, except for the members themselves, are composite) is ten, given by

for , 1, ..., 5 (Lander and Parkin 1967, Dubner and Nelson 1997). According to Dubner

It is conjectured that there are arbitrarily long sequences of Primes in Arithmetic Progression (Guy 1994).

**References**

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Lander, L. J. and Parkin, T. R. ``Consecutive Primes in Arithmetic Progression.'' *Math. Comput.* **21**, 489, 1967.

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Pritchard, P. A.; Moran, A.; and Thyssen, A. ``Twenty-Two Primes in Arithmetic Progression.'' *Math. Comput.*
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Zimmerman, P. http://www.loria.fr/~zimmerma/records/8primes.announce.

© 1996-9

1999-05-26