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A027206
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Numbers m such that (1+i)^m + i is a Gaussian prime.
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3
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0, 1, 2, 3, 4, 6, 7, 8, 11, 14, 16, 19, 38, 47, 62, 79, 151, 163, 167, 214, 239, 254, 283, 367, 379, 1214, 1367, 2558, 4406, 8846, 14699, 49207, 77291, 160423, 172486, 221006, 432182, 1513678, 2515574
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OFFSET
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1,3
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COMMENTS
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Equivalently, either (1+i)^m + i times its conjugate is an ordinary prime, or m == 2 (mod 4) and 2^(m/2) + (-1)^((m-2)/4) is an ordinary prime.
Let z = (1+i)^m + i. If z is not pure real or pure imaginary, then z is a Gaussian prime if the product of z and its conjugate is a rational prime. That product is 1 + 2^m + sin(m*Pi/4)*2^(1+m/2). z is imaginary when m=4k+2, in which case z has magnitude 2^(2k+1) + (-1)^k. These pure imaginary numbers are Gaussian primes when 2^(2k+1)-1 is a Mersenne prime and 2k+1 == 1 (mod 4); that is, when m is twice an odd number in A112633. - T. D. Noe, Mar 07 2011
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LINKS
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MATHEMATICA
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Select[Range[0, 30000], PrimeQ[(1+I)^#+I, GaussianIntegers->True]&]
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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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