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A033235
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Primes of the form x^2 + 55*y^2.
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3
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59, 71, 199, 229, 251, 269, 311, 379, 389, 499, 509, 631, 661, 691, 751, 839, 881, 929, 1049, 1061, 1171, 1181, 1279, 1321, 1409, 1439, 1499, 1571, 1609, 1699, 1721, 1741, 1901, 1951, 2029, 2069, 2269
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OFFSET
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1,1
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COMMENTS
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Also primes of the form x^2 - xy + 14y^2 with x and y nonnegative. - T. D. Noe, May 08 2005
Conjecture: consecutive elements of this sequence are consecutive primes satisfying the congruence b(k) == 1 (mod k) for k>0, where b(k) is recursive sequence defined as follows: b(k) = -b(k-1) - b(k-2) + b(k-3) - b(k-4) with b(0)=2, b(1)=1, b(2)=0, b(3)=-1.
(b(59) - 1) mod 59 = (-496870918 - 1) mod 59 = 0, 59 = a(1).
(b(71) - 1) mod 71 = (88081764473 - 1) mod 71 = 0, 71 = a(2).
For 10^6 consecutive positive integers there are 9748 prime solutions and 5 nonprime (1, 586, 2935, 17161, 429737) solutions of the congruence. (End)
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REFERENCES
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David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
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LINKS
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MATHEMATICA
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QuadPrimes2[1, 0, 55, 10000] (* see A106856 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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