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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, 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

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

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Last modified September 25 21:49 EDT 2022. Contains 356986 sequences. (Running on oeis4.)