NUMERICAL RESULTS ON BEST UNIFORM RATIONAL APPROXIMATION OF VERTICAL-BAR-X-VERTICAL-BAR ON [-1, +1]

With E(n,n)(Absolute value of x; [-1, +1]) denoting the error of best uniform rational approximation from pi(n,n) to Absolute value of x on [-1, +1], we determine the numbers {E2n,2n(Absolute value of x; [-1, 1])}n=1(40), where each of these numbers was calculated with a precisi on of at least 200 significant digits. With these numbers, the Richard son extrapolation method was applied to the products {e(pi square-root 2n)E2n,2n(Absolute value of x; [-1, +1])}n=1(40) and it appears, to a t least 10 significant digits, that 8=?lim(n-->infinity)e(pi square-ro ot 2n)E2n,2n(Absolute value of x; [-1, +1] which gives rise to an inte resting new conjecture in the theory of rational approximation.