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A057619
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Initial prime in first sequence of n primes congruent to 3 modulo 4.
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6
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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
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OFFSET
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1,1
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COMMENTS
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The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017
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REFERENCES
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R. K. Guy, "Unsolved Problems in Number Theory", A4
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LINKS
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EXAMPLE
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a(13) = 241603 because this number is the first in a sequence of 13 consecutive primes all of the form 4n + 3.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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