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A342368
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Fundamental discriminants of real quadratic number fields with odd class number > 1.
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3
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229, 257, 316, 321, 401, 469, 473, 568, 577, 733, 761, 817, 892, 993, 1009, 1016, 1093, 1101, 1129, 1229, 1257, 1297, 1304, 1373, 1393, 1429, 1436, 1489, 1509, 1601, 1641, 1756, 1761, 1772, 1897, 1901, 1929, 1957, 1996, 2021, 2029, 2081, 2089, 2101, 2153, 2177, 2213
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OFFSET
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1,1
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COMMENTS
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For a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. See Theorem 1 and Theorem 2 of Ezra Brown's link. A003656 gives the case where the class number is 1.
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LINKS
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EXAMPLE
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The class number of the quadratic field with discriminant 229 (namely Q(sqrt(229)) is 3, so 229 is a term.
The class number of the quadratic field with discriminant 1756 (namely Q(sqrt(439)) is 5, so 1756 is a term.
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PROG
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(PARI) isA342368(D) = if((D>1) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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