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

Author

Steven Finch, Jun 13 2005

Keywords

• nonn

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

Crossrefs

Cf. A107998.

Programs

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 *)