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A349419
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Fundamental discriminants of real quadratic number fields with odd class number > 1 whose fundamental unit has norm 1.
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1
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316, 321, 469, 473, 568, 817, 892, 993, 1016, 1101, 1257, 1304, 1393, 1436, 1509, 1641, 1756, 1761, 1772, 1897, 1929, 1957, 1996, 2021, 2101, 2177, 2429, 2589, 2636, 2908, 2913, 2981, 3173, 3261, 3356, 3569, 3736, 3873, 3928, 3941, 3957, 3981, 3997, 4009, 4193, 4281
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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. This sequence gives values for d in the cases (ii) and (iii) and that the real quadratic number field with discriminant d has odd class number > 1.
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LINKS
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EXAMPLE
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316 is a term since the quadratic field with discriminant 316 (Q(sqrt(79)) has class number 3. The fundamental unit of that field (80+9*sqrt(79)) has norm 1.
321 is a term since the quadratic field with discriminant 321 (Q(sqrt(321)) has class number 3. The fundamental unit of that field (215+12*sqrt(321)) has norm 1.
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PROG
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(PARI) isA349419(D) = if(!isprime(D) && (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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