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

5, 8, 10, 12, 13, 15, 17, 18, 20, 21, 24, 26, 27, 28, 29, 30, 32, 33, 35, 37, 39, 40, 41, 42, 44, 45, 48, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 65, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 84, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99, 101, 103

Offset

1,1

Comments

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.

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.

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

Links

Table of n, a(n) for n=1..67.

R. Dedekind, Sur la théorie des nombres entiers algébriques, Gauthier-Villars, 1877.

Mathematiques.net, Anneaux factoriels

Encyclopedia of Mathematics, Factorial ring

Dany-Jack Mercier, Anneaux factoriels, 2003. In French.

R. Raghavendran, Finite associative rings, Compositio Mathematica, vol 21, no 2 (1969) pp. 195-229.

Formula

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.

Example

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

Mathematica

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 *)

Crossrefs

Cf. A003173, A003172.

Sequence in context: A049195 A172019 A064362 * A248356 A115401 A314377

Adjacent sequences: A173295 A173296 A173297 * A173299 A173300 A173301

Keyword

nonn

Author

Michel Lagneau, Feb 15 2010; corrected Feb 22 2010

Extensions

Incorrect term 94 removed by Michel Lagneau, Dec 18 2018

Status

approved