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A341649
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Integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a unique factorization domain.
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1
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-2, -1, 2, 3, 6, 7, 11, 14, 19, 22, 23, 31, 38, 43, 46, 47, 59, 62, 67, 71, 83, 86, 94, 103, 107, 118, 127, 131, 134, 139, 151, 158, 163, 166, 167, 179, 191, 199, 206, 211, 214, 227, 239, 251, 262, 263, 271, 278, 283, 302, 307, 311, 331, 334, 347, 358, 367, 379
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OFFSET
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1,1
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COMMENTS
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Equivalently, integers k such that Z[sqrt(k)] = Z[x]/(x^2 - k) is a principal ideal domain.
-2, -1, together with k such that 4*k is in A003656.
All terms are squarefree and congruent to 2 or 3 modulo 4. It appears that the terms > 2 are of the form p or 2*p, where p is a prime congruent to 3 modulo 4. [This is correct; see Theorem 1 and Theorem 2 of Ezra Brown's link. - Jianing Song, Feb 24 2021]
The smallest prime p == 3 (mod 4) that is not a term is p = 79. The smallest prime p == 3 (mod 4) such that 2*p is not a term is p = 71.
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LINKS
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EXAMPLE
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Z[sqrt(-1)] = Z[i] is the ring of Gaussian integers, which is a unique factorization domain.
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PROG
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(PARI) isA341649(n) = my(D=4*n); isfundamental(D) && quadclassunit(D)[1] == 1
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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