It's easy to find the best fractional approximations for the square root of 2, based on the simple continued fraction. This gives convergents 7/5, 17/12, 41/29, and so on. However, it's not so easy to define the analagous sequence for CUBE root of 2. Lagrange proved the simple continued fraction of an irrational number is periodic if and only if the number is a quadratic irrational, i.e., the root of a quadratic equation, such as x^2 - 2 = 0. As a result, the numerators and denominators of successive convergents of the continued fraction for sqrt(N) can always be generated by a simple second order linear recurrence. This corresponds to the solutions of the basic "Pell equation" x^2 - Ny^2 = 1. On the other hand, solutions of the analagous equation x^3 - Ny^3 = 1 for cubes are less well-behaved. Nevertheless, it's always possible to construct a linear recurring sequence of integers s[0], s[1], s[2],... such that any specified algebraic number is approached by some function of the ratio of successive terms s[n+1]/s[n]. In particular, to construct a sequence for the rth root of N, let m denote the largest integer such that m^r < N. Then we have the rth order recurrence relation r / r \ (r-j) (j-1) s[n] = SUM ( ) m (N-m^r) s[n-j] j=1 \ j / with the initial values s[0]=s[1]=..s[r-2]=0 and s[r-1]=1, any by construction we have s[n] N^(1/r) = lim m + (N-m^r) ------ n->inf s[n+1] For example, to construct the sequence for the cube root of 2 we have N=2 and r=3, so m=1 and the recurrence is s[n] = 3 s[n-1] + 3 s[n-2] + s[n-3] The approximations for 2^(1/3) given by successive ratios of terms in this sequence are listed below, along side the convergents of the continued fraction generated by convergents of linear recurrence continued fraction 1/|N(N-DV)| 4/3 * 4/3 1.135 5/4 * 5/4 5.040 29/23 * 29/23 1.581 223/177 34/27 1.646 63/50 4.021 286/227 * 286/227 1.682 3301/2620 349/277 1.533 635/504 * 635/504 7.530 5429/4309 * 5429/4309 0.940 6064/4813 * 6064/4813 12.546 723235/574032 90325/71691 0.924 463753/368081 96389/76504 8.964 10705243/8496757 1054215/836731 2.298 957826/760227 2204819/1749966 1.368 52819267/41922680 3259034/2586697 3.787 15240955/12096754 * 15240955/12096754 8.806 2345474521/186104361 186150494/147747745 1.834 Those marked with an asterisk are the same on both lists. This raises the interesting question of whether infinitely many convergents occur in the sequence generated by the 3rd order recurrence. We might also wonder which of the above approximations is the "best". Obviously the absolute accuracy improves indefinitely, but we can define a notion of "goodness" by considering the ratio of how many digits of accuracy are achieved per digit of the numerator and denominator. One such measure for the approximation N/D to the value V is 1/|N(N-DV)|. The goodness of each convergent is listed in the right hand column of the table above. It's interesting that the goodness of the continued fractions never falls much below 1.0, whereas the linear recurrence approximations can have very low values of "goodness". For example, the goodness of 723235/574032 is just 0.0122. I wonder what can be said about the distribution of "goodness" of continued fraction convergents. For example, how soon would we expect a convergent with more goodness than 12.546, which is the maximum for all the values listed in the table?

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