Square Roots with "All" 2's in Periodic Continued Fraction

Len Smiley, Univ. of Alaska Anchorage

m=Floor(Sqrt(m))Period of Continued Fraction for Sqrt(m)
n2+3n+2 [n+1] [2,2n+2]
25n2+14n+2 [5n+1] [2,2,10n+2]
36n2+17n+2 [6n+1] [2,2,2,12n+2]
841n2+82n+2 [29n+1] [2,2,2,2,58n+2]
1225n2+99n+2 [35n+1] [2,2,2,2,2,70n+2]

Example: Row 2 with n=2: the continued fraction expansion for the square root of 130 is [11,2,2,22,2,2,22,2,2,22,...].

Unlike the "all 1's" case, we can find an infinite family of integers m with the palindromic part of the period in the continued fraction expansion of the square root of m consisting of ANY number, k, of 2's. We prefer, however, to write two formulas for m: one if k is even:

m = Pk+12 n2 + 2 ( Pk+1 + Pk) n + 2

and one when k is odd:

m = (Pk+12/4) n2 + ( Pk+1 + Pk) n + 2

where Pk is the k-th Pell number. See also Sloane's A000129.

Len M. Smiley
Last modified: Mon Jul 17 18:56:35 AKDT 2000