A172000 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm -1.

2, 5, 8, 10, 13, 17, 18, 20, 26, 29, 32, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 137, 145, 148, 149, 153, 157, 160, 162, 164, 170, 173, 180, 181, 185, 193, 197, 200

Offset

1,1

Comments

Complement of A087643 in the nonsquare integers A000037.

Subsequence of A000415, their set difference form A172001.

Contains A003814 as a subsequence, their squarefree terms coincide and form A003654.

It seems that this sequence also gives the values of n such that the equation x^2 - n*y^2 = n has integer solutions. - Colin Barker, Aug 20 2013

Links

Table of n, a(n) for n=1..60.

Formula

A positive integer n is in this sequence iff its squarefree core A007913(n) belongs to A003654.

Mathematica

cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d3 = Expand[(d1 + d2) (d1 - d2)]; If[d3 == -1, AppendTo[cr, n]]], {n, 2, 1000}]; cr (* Artur Jasinski, Oct 10 2011 *)

Prog

(PARI) { for(n=1, 1000, if(issquare(n), next); if( norm(bnfinit(x^2-n).fu[1])==-1, print1(n, ", ")) ) }

Crossrefs

Sequence in context: A025284 A140328 A000415 * A096691 A202057 A263651

Adjacent sequences: A171997 A171998 A171999 * A172001 A172002 A172003

Keyword

nonn

Author

Max Alekseyev, Jan 21 2010

Extensions

Edited by Max Alekseyev, Mar 09 2010

Status

approved