A305273 - OEIS

%I #40 Dec 11 2019 08:55:20

%S -11,-7,-3,5,13,17,21,29,33,37,41,57,73

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

%C 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:

%C a) d <= -3,

%C b) d == 1 (mod 4),

%C c) d has a square divisor different of 1,

%C d) the number 2 is irreducible in Z[sqrt(d)]. Consequently, the equation x^2 - dy^2 = -2 or +2 has no solution.

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

%D 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.

%D H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 294.

%H Encyclopedia of Mathematics, <a href="https://www.encyclopediaofmath.org/index.php/Factorial_ring">Factorial ring</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Unique_factorization_domain">Unique factorization domain</a>

%e 5 = A048981(8) is in the sequence because the squarefree number 5 == 1 (mod 4) implies that Z[sqrt(5)] is not UFD.

%e 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.

%p with(numtheory):T:=array(1..18):

%p A048981:=[-11, -7, -3, -2, -1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 ]:

%p for n from 1 to 21 do:

%p if A048981[n]<=-3

%p    or issqrfree(A048981[n])=false

%p    or irem(A048981[n],4)=1

%p    or nops(factorEQ(2, A048981[n]))=1

%p    then

%p    printf(`%d, `,A048981[n]):

%p    else

%p fi:

%p od:

%Y Cf. A003174, A048981, A173298.

%K sign,fini,full

%O 1,1

%A _Michel Lagneau_, Dec 17 2018