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A002052
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Prime determinants of forms with class number 2.
(Formerly M4339 N1816)
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3
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3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 367, 379, 383, 419
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OFFSET
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1,1
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COMMENTS
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The Suryanarayana paper contains these errors: In section 2, list (1) omits 3 and an asterisk should follow 1987; list (2) should include neither 3203 nor 3271. Section 3 should say "Of the 339 primes d == 3 (4) up to 5000, 289 primes satisfy h(d) = 2, while 50 do not." (correcting all three counts) - Rick L. Shepherd, Apr 29 2015
Also primes p > 2 such that Z[sqrt(p)] = Z[x]/(x^2 - p) is a unique factorization domain (or equivalently, a principal ideal domain). This can be deduced from the following result: let K be the quadratic field with discriminant D > 0, h(D) and h_+(D) be the ordinary class number and narrow class numer (or form class number) of K respectively, then h_+(D)/h(D) = 1 if the fundamental unit of K has norm -1; 2 if the fundamental unit of K has norm 1. - Jianing Song, Feb 17 2021
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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PROG
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(PARI) {QFBclassno(D) = qfbclassno(D) * if(D < 0 || norm(quadunit(D)) < 0, 1, 2);
n=0; forprime(p=3, 291619, if(p%4 == 3 && QFBclassno(4*p) == 2, n++; write("b002052.txt", n, " ", p)))} \\ Rick L. Shepherd, Apr 29 2015
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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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