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A173298 Integers n >= 2 such that the ring Z(sqrt n) is not factorial. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

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Last modified December 1 11:14 EST 2022. Contains 358468 sequences. (Running on oeis4.)