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A296926
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Rational primes that decompose in the field Q(sqrt(-13)).
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5
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7, 11, 17, 19, 29, 31, 47, 53, 59, 61, 67, 71, 83, 101, 113, 151, 157, 163, 167, 173, 181, 223, 227, 233, 239, 257, 269, 271, 277, 307, 313, 331, 337, 359, 373, 379, 383, 389, 431, 433, 463, 479, 487, 499, 521, 569, 587, 601, 619, 631, 641, 643, 653, 673, 677, 683, 691
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OFFSET
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1,1
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COMMENTS
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In general, primes that decompose in Q(sqrt(-p prime)) are congruent modulo 4p to t(-1)^[t^(phi(p)/2) mod p = 1 XOR t mod min(e,4) = 1], where t are the totatives of 2p, e is the even part of phi(p), and [P] returns 1 if P else 0. In other words, if phi(p) is at least twice even, then the t are signed so that the quadratic residuosity of t mod p aligns with the congruence of +-t mod 4 to 1--the modulus 4p is thence irreducible--; if only once, then the signature simply indicates quadratic residues modulo p. The imbalance of signs in either flank (t < p, t > p) of the signature also gives the class number of Q(sqrt(-p)), up to an excess factor of 3 if p == 3 (mod 8) but != 3. [E.g., for p = 13 we have +--+++ or +++--+, so the class number of Q(sqrt(-13)) = 2; for p = 11 == 3 (mod 8) we have +++-+ or -+---, so the class number of Q(sqrt(-11)) = 3/3 = 1.] - Travis Scott, Jan 05 2023
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LINKS
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FORMULA
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Primes == {1, 7, 9, 11, 15, 17, 19, 25, 29, 31, 47, 49} (mod 52). - Travis Scott, Jan 05 2023
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MAPLE
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MATHEMATICA
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Select[Prime[Range[125]], KroneckerSymbol[-13, #] == 1 &] (* Amiram Eldar, Nov 17 2023 *)
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PROG
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(PARI) list(lim)=my(v=List()); forprime(p=5, lim, if(kronecker(-13, p)==1, listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Mar 18 2018
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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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