A088430 a(n) = the least positive d such that for p=prime(n), the numbers p+0d, p+1d, p+2d, ..., p+(p-1)d are all primes.

1, 2, 6, 150, 1536160080, 9918821194590, 341976204789992332560, 2166703103992332274919550

Offset: 1

Author

Zak Seidov, Sep 30 2003

Keywords

• more
• nonn

Comments

Problem discussed by Russell E. Rierson: starting with given p, find the least d such that the arithmetic progression p,p+d,p+2d,... contains only primes. Obviously, the maximum number of prime terms is p and to reach that maximum, d must be a multiple of all smaller primes. For example, a(5) is a multiple of 2*3*5*7.

There can be other maximum-length prime progressions starting at p, with larger d. (Zak Seidov found d=4911773580 for p=11.)

Examples

			n AP Last term
--------------
1 2+i 3
2 3+2*i 7
3 5+6*i 29
4 7+150*i 907
5 11+1536160080*i 15361600811
6 13+9918821194590*i 119025854335093
7 17+341976204789992332560*i 5471619276639877320977
8 19+2166703103992332274919550*i 39000655871861980948551919
		

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 139-140.

Links

• Jens Kruse Andersen, Smallest AP-k with minimal start
• Phil Carmody, a(7), NMBRTHRY Nov 2001
• Andrew Granville, Prime number patterns
• Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, arXiv:math/0404188 [math.NT], 2004-2007. [Background]
• Paulo Ribenboim, Les records des nombres premiers, Séminaire de Philosophie et Mathématiques (8) (1987) 1-25.
• Paulo Ribenboim, Prime number records, Coll. Math. J. 25 (4) (1994) 280-290.
• Paulo Ribenboim, Euler's Famous prime generating polynomial and the class number of imaginary quadratic fields, (2000) p 91-111
• Russell E. Rierson, Question About Prime Numbers.
• Zak Seidov, Question About Prime Numbers.
• Zak Seidov and others, Russell E. Rierson's Question About Prime Numbers, digest of 5 messages in primenumbers Yahoo group, Sep 29 - Oct 1, 2003.

Crossrefs

See A113834 for last term in the progression, and A231017 for the 2nd term.

Cf. A061558, A231018, A002110.

Programs

Mathematica

A088430[n_] := Module[{p, m, d},
   p = Prime[n]; m = Product[Prime[i], {i, 1, n - 1}];
   d = m;
   While[! AllTrue[Table[p + i*d, {i, 1, p - 1}], PrimeQ], d = d + m];
   Return[d];
   ];
Table[A088430[n], {n, 1, 8}] (* Robert Price, Mar 27 2019 *)

Formula

a(n) = A231017(n) - prime(n). - Jonathan Sondow, Nov 08 2013

a(n) = A061558(prime(n)). - Jens Kruse Andersen, Jun 30 2014

a(n) = A002110(n-1) * A231018(n). - Jeppe Stig Nielsen, Mar 16 2016

Extensions

Edited by Don Reble, Oct 04 2003

a(7) was found by Phil Carmody. - Don Reble, Nov 23 2003

Entry revised by N. J. A. Sloane, Jan 25 2006

a(8) found by Wojciech Izykowski. - Jens Kruse Andersen, Jun 30 2014