A107997 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y odd.

5, 13, 21, 29, 53, 61, 69, 77, 85, 93, 109, 133, 149, 157, 165, 173, 181, 205, 213, 221, 229, 237, 253, 277, 285, 293, 301, 309, 317, 341, 357, 365, 397, 413, 421, 429, 437, 445, 453, 461, 469, 493, 501, 509, 517, 533, 541, 565, 581, 589, 597, 613, 629, 645

Offset

1,1

Comments

Squarefree integers m for which the fundamental unit of Q(sqrt(m)) is of the form (u + v*sqrt(m))/2, where u and v are both odd.

References

E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields, British Association Mathematical Tables, Vol. IV, London, 1934.

H. C. Williams, Eisenstein's problem and continued fractions, Utilitas Math. 37 (1990) 145-157.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 161 terms from Charles R Greathouse IV)

F. Arndt, Beiträge zur Theorie der quadratischen Formen, Archiv der Mathematik und Physik 15 (1850) 467-478.

A. Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod 8), J. Reine Angew. Math. 53 (1857) 369-371.

Steven R. Finch, Class number theory

Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Eric Weisstein's World of Mathematics, Fundamental unit

Mathematica

fQ[n_] := Block[{nffu = NumberFieldFundamentalUnits@ Sqrt@ n}, SquareFreeQ@ n && Denominator[ nffu[[1, 2, 2]]] > 1]; Select[ 8Range@ 81 - 3, fQ] (* Robert G. Wilson v, Dec 22 2014 *)

Crossrefs

Cf. A107998.

Sequence in context: A004770 A191155 A107996 * A355461 A166095 A166090

Adjacent sequences: A107994 A107995 A107996 * A107998 A107999 A108000

Keyword

nonn

Author

Steven Finch, Jun 13 2005

Status

approved