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A173298 Integers n >= 2 such that the ring Z(sqrt n) is not factorial. 1

%I #37 May 01 2023 10:02:41

%S 5,8,10,12,13,15,17,18,20,21,24,26,27,28,29,30,32,33,35,37,39,40,41,

%T 42,44,45,48,50,52,53,54,55,56,57,58,60,61,63,65,68,69,70,72,73,74,75,

%U 76,77,78,80,82,84,85,87,88,89,90,91,92,93,95,96,97,98,99,101,103

%N Integers n >= 2 such that the ring Z(sqrt n) is not factorial.

%C A factorial ring 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 Z(sqrt n), where n >=2 such that this ring is not factorial. It is well known that the ring Z(sqrt n) is not factorial if it satisfies the following conditions: n == 1 mod 4, n has a square divisor different of 1 and the number 2 is irreducible in Z(sqrt n). In consequence, the equation x^2 - ny^2 = -2 or +2 has no solution.

%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 W. Krull, Idealtheorie, Springer Verlag, 1937 (2e edition 1968)

%H R. Dedekind, <a href="http://www.numdam.org/item?id=BSMA_1876__11__278_0">Sur la théorie des nombres entiers algébriques</a>, Gauthier-Villars, 1877.

%H Mathematiques.net, <a href="http://www.les-mathematiques.net/p/g/b/node8.php">Anneaux factoriels</a>

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

%H Dany-Jack Mercier, <a href="https://megamaths.fr/cours/ari/cann0001.pdf">Anneaux factoriels</a>, 2003. In French.

%H R. Raghavendran, <a href="http://www.numdam.org/item?id=CM_1969__21_2_195_0">Finite associative rings</a>, Compositio Mathematica, vol 21, no 2 (1969) pp. 195-229.

%F We calculate n from the conditions : n == 1 mod. 4, or n has a square integer which divides n, or the equation x^2 - ny^2 = -2 or +2 has no solution.

%e For n = 3, n == 3 (mod 4) and no square divide 3. 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.

%e For n = 5, n == 1 (mod 4), the ring Z(sqrt 5) is not factorial.

%e For n = 87, n == 3 (mod 4) and no square divide 87, but the equation x^2 - 87y^2 = -2 or +2 has no solution. The ring Z(sqrt 87) is not factorial.

%t lst={};Do[ii=0;Do[If[IntegerQ[Sqrt[n*x^2+2]]||IntegerQ[Sqrt[n*x^2-2]],ii=1],{x,2,10^5}];If[!IntegerQ[Sqrt[n]]&&(ii==0||Mod[n,4]==1||!SquareFreeQ[n]),AppendTo[lst,n]],{n,2,100}];lst (* _Michel Lagneau_, Dec 18 2018 *)

%Y Cf. A003173, A003172.

%K nonn

%O 1,1

%A _Michel Lagneau_, Feb 15 2010; corrected Feb 22 2010

%E Incorrect term 94 removed by _Michel Lagneau_, Dec 18 2018

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)