A197170 Smallest k such that the fundamental unit (x+y*w) or (x+y*w)/2 of the real quadratic field Q(sqrt(k)) obeys gcd(k,y)=n.

6, 69, 248, 115, 78, 511, 1016, 603, 70, 385, 3432, 793, 238, 2655, 14224, 1241, 3186, 703, 3980, 9177, 154, 736, 456, 1825, 3172, 13959, 2884, 319, 1110, 4619, 7136, 10659, 7174, 10255, 44856, 7067, 2926, 16185, 54280, 779, 7602, 10879, 22088, 10215, 46

Offset: 2

Author

Artur Jasinski, Oct 11 2011

Keywords

• nonn

Comments

Conjecture: For every n such a quadratic field with minimum k exists.

Examples

			For n=2 the unit is 2*w-5 with k=6.
For n=3 the unit is (3*w+25)/2 with k=69.
For n=4 the unit is (4*w-63) with k=248.
For n=5 the unit is 105*w-1126 with k=115.
For n=7 the unit is 185290497*w-4188548960 with k=511 (and this x and y appear in A041976 and A041977).
		

Crossrefs

Cf. A087643, A172000, A194366, A197115, A197127, A197128.

Programs

Mathematica

cr = {}; ck = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[ck, GCD[d4, n]];  AppendTo[cr, n]]], {n, 2, 200000}]; aa = {}; Do[AppendTo[aa, cr[[First[Position[ck, n]][[1]]]]], {n, 2, 99}]; aa

Formula

k = A197127(m) where m is the smallest m such that A197169(m)=n.