

A305273


Numbers k in A048981 for which the ring Z[sqrt(k)] is not a UFD.


0



11, 7, 3, 5, 13, 17, 21, 29, 33, 37, 41, 57, 73
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.
So the ring Z[A048981(n)] is factorial for the following values of A048981: 2, 1, 2, 3, 6, 7, 11 and 19.


REFERENCES

R. Dedekind, Sur la théorie des nombres entiers algébriques, GauthierVillars, 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

Table of n, a(n) for n=1..13.
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

Cf. A003174, A048981, A173298.
Sequence in context: A109828 A048981 A132361 * A236546 A155914 A087896
Adjacent sequences: A305270 A305271 A305272 * A305274 A305275 A305276


KEYWORD

sign,fini,full


AUTHOR

Michel Lagneau, Dec 17 2018


STATUS

approved



