OFFSET
1,1
COMMENTS
The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017
REFERENCES
R. K. Guy, "Unsolved Problems in Number Theory", A4
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..36 (terms < 4*10^14)
J. K. Andersen, Consecutive Congruent Primes.
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]
EXAMPLE
a(13) = 241603 because this number is the first in a sequence of 13 consecutive primes all of the form 4n + 3.
MATHEMATICA
NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; Return[ k ] ]; p = 0; Do[ a = Table[ -1, {n} ]; k = Max[ 1, p ]; While[ Union[ a ] != {3}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 4 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 18} ]
With[{prs=Table[If[Mod[Prime[n], 4]==3, 1, 0], {n, 4646*10^6}]}, Prime/@ Table[ SequencePosition[prs, PadRight[{}, k, 1], 1][[1, 1]], {k, 26}]] (* The program will take a long time to run and requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 28 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Oct 09 2000
EXTENSIONS
More terms from Don Reble, Nov 16 2003
More terms from Jens Kruse Andersen, May 29 2006
STATUS
approved