A057624 Initial prime in first sequence of n primes congruent to 1 modulo 4.

5, 13, 89, 389, 2593, 11593, 11593, 11593, 11593, 373649, 766261, 3358169, 12204889, 12270077, 12270077, 12270077, 297387757, 297779117, 297779117, 1113443017, 1113443017, 1113443017, 1113443017, 1113443017, 84676452781, 84676452781, 689101181569, 689101181569, 689101181569, 3278744415797, 3278744415797, 3278744415797, 3278744415797

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(9) = 11593 because "[t]his number is the first in a sequence of 9 consecutive primes all of the form 4n + 1."
		

References

• R. K. Guy, Unsolved Problems in Number Theory, A4.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 163.

Links

• Jens Kruse Andersen, Consecutive Congruent Primes.
• D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Crossrefs

Cf. A057619, A057620, A057622, A055623, A055624.

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 ] != {1}, k = NextPrime[ k ]; a = Take[ AppendTo[ a, Mod[ k, 4 ] ], -n ] ]; p = NestList[ PrevPrime, k, n ]; Print[ p[ [ -2 ] ] ]; p = p[ [ -1 ] ], {n, 1, 19} ]

Extensions

More terms from Don Reble, Nov 16 2003

More terms from Jens Kruse Andersen, May 29 2006