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A305273
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Numbers k in A048981 for which the ring Z[sqrt(k)] is not a UFD.
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0
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-11, -7, -3, 5, 13, 17, 21, 29, 33, 37, 41, 57, 73
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graph;
refs;
listen;
history;
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internal format)
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OFFSET
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1,1
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COMMENTS
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A factorial ring (or UFD = unique factorization domain) is an integral domain in which one can find a system of irreducible elements P such that every nonzero element admits a unique representation. We consider the ring A = Z[A048981(n)] such that A is not a UFD for some n. In the general case, it is well known that Z[sqrt(d)] is not factorial if one of the following conditions is satisfied:
a) d <= -3,
b) d == 1 (mod 4),
c) d has a square divisor different of 1,
d) the number 2 is irreducible in Z[sqrt(d)]. Consequently, the equation x^2 - dy^2 = -2 or +2 has no solution.
So the ring Z[A048981(n)] is factorial for the following values of A048981: -2, -1, 2, 3, 6, 7, 11 and 19.
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REFERENCES
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R. Dedekind, Sur la théorie des nombres entiers algébriques, Gauthier-Villars, 1877. English translation with an introduction by J. Stillwell: Theory of Algebraic Integers, Cambridge Univ. Press, 1996.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 294.
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LINKS
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EXAMPLE
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5 = A048981(8) is in the sequence because the squarefree number 5 == 1 (mod 4) implies that Z[sqrt(5)] is not UFD.
3 = A048981(7) is not in the sequence because the squarefree number 3 is not congruent to 1 (mod 4), but the solutions of the equation x^2 - 3y^2 = -2 or +2 are x = 1 (or -1), y = 1 (or -1). The ring Z[sqrt(3)] is factorial.
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MAPLE
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with(numtheory):T:=array(1..18):
A048981:=[-11, -7, -3, -2, -1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 ]:
for n from 1 to 21 do:
then
else
fi:
od:
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CROSSREFS
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KEYWORD
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sign,fini,full
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AUTHOR
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STATUS
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approved
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