A057619 Initial prime in first sequence of n primes congruent to 3 modulo 4.

3, 7, 199, 199, 463, 463, 463, 36551, 39607, 183091, 241603, 241603, 241603, 9177431, 9177431, 95949311, 105639091, 341118307, 727334879, 727334879, 1786054147, 1786054147, 22964264027, 54870713243, 79263248027, 113391385603

Offset: 1

Author

Robert G. Wilson v, Oct 09 2000

Keywords

• nonn

Comments

The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017

Examples

			a(13) = 241603 because this number is the first in a sequence of 13 consecutive primes all of the form 4n + 3.
		

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]

Crossrefs

Cf. A057620, A057622, A057624.

Programs

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 *)

Extensions

More terms from Don Reble, Nov 16 2003

More terms from Jens Kruse Andersen, May 29 2006