OFFSET
1,1
COMMENTS
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.
REFERENCES
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.
LINKS
Encyclopedia of Mathematics, Factorial ring
Wikipedia, Unique factorization domain
EXAMPLE
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.
MAPLE
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:
if A048981[n]<=-3
or issqrfree(A048981[n])=false
or irem(A048981[n], 4)=1
or nops(factorEQ(2, A048981[n]))=1
then
printf(`%d, `, A048981[n]):
else
fi:
od:
CROSSREFS
KEYWORD
sign,fini,full
AUTHOR
Michel Lagneau, Dec 17 2018
STATUS
approved