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 A185720 Square roots of discriminants of norm-Euclidean Galois cubic fields. 1
 7, 9, 13, 19, 31, 37, 43, 61, 67, 103, 109, 127, 157 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Heilbronn shows that this sequence is finite. McGown 2010 strengthens that result, showing that the largest term is less than 10^70. Following Godwin & Smith, Lemmermeyer showed that there are no further terms below 500,000. Theorem 1.1 of McGown 2011: Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant Delta = 7^2, 9^2, 13^2, 19^2, 31^2, 37^2, 43^2, 61^2, 67^2, 103^2, 109^2, 127^2, 157^2. LINKS Table of n, a(n) for n=1..13. H. J. Godwin and J. R. Smith, On the Euclidean nature of four cyclic cubic fields, Math. Comp. 60:201 (1993), pp. 421-423. H. Heilbronn, On Euclid's algorithm in cyclic fields, Canadian J. Math. 3 (1951), pp. 257-268. Franz Lemmermeyer, The Euclidean algorithm in algebraic number fields, 2004. Kevin J. McGown, Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis, arXiv:1102.2043 [math.NT], 2011. Kevin J. McGown, Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis, Journal de Théorie des Nombres de Bordeaux, 24 (2012), 425-445. Kevin J. McGown, Norm-Euclidean Galois fields, arXiv:1011.4501 [math.NT], 2010-2011. EXAMPLE a(10)^2 = 24649 = 157^2. CROSSREFS Cf. A048981 Squarefree values of n for which the quadratic field Q[sqrt(n)] is Euclidean. Cf. A297063. Sequence in context: A027692 A343001 A297063 * A032487 A332103 A160777 Adjacent sequences: A185717 A185718 A185719 * A185721 A185722 A185723 KEYWORD nonn,fini AUTHOR Jonathan Vos Post and Charles R Greathouse IV, Feb 10 2011 EXTENSIONS 31, 37, and 43 from Robert C. Lyons, Dec 25 2017 STATUS approved

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Last modified September 8 00:04 EDT 2024. Contains 375749 sequences. (Running on oeis4.)