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A055673
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Absolute values of norms of primes in ring of integers Z[sqrt(2)].
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11
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2, 7, 9, 17, 23, 25, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 121, 127, 137, 151, 167, 169, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 361, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593
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OFFSET
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1,1
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COMMENTS
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The integers have the form z = a + b*sqrt(2), a and b rational integers. The norm of z is a^2 - 2*b^2, which may be negative.
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REFERENCES
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L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VII.
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LINKS
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FORMULA
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Consists of 2; rational primes = +-1 (mod 8); and squares of rational primes = +-3 (mod 8).
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MATHEMATICA
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maxNorm = 593; s1 = Select[Range[-1, maxNorm, 8], PrimeQ]; s2 = Select[Range[1, maxNorm, 8], PrimeQ]; s3 = Select[Range[-3, Sqrt[maxNorm], 8], PrimeQ]^2; s4 = Select[Range[3, Sqrt[maxNorm], 8], PrimeQ]^2; Union[{2}, s1, s2, s3, s4] (* Jean-François Alcover, Dec 07 2012, from formula *)
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PROG
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(PARI) is(n)=!!if(isprime(n), setsearch([1, 2, 7], n%8), issquare(n, &n) && isprime(n) && setsearch([3, 5], n%8)) \\ Charles R Greathouse IV, Sep 10 2016
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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I would also like to get the sequences (analogous to A055027 and A055029) giving the number of inequivalent primes mod units. Of course now there are infinitely many units.
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STATUS
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approved
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