The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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