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
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
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