Bill Gosper <billgosper@gmail.com> wrote:
Off-list I grumbled that no self-respecting Deity would bother sending clues to worshipers dumb enough to use decimal instead of continued fractions.
Perhaps there's a hidden message in the continued fraction for e. :-) Why not Egyptian fractions? The Egyptians, ancient and modern, have always been very religious. Any positive real number can be approximated to any desired precision by an Egyptian fraction, i.e. the sum of reciprocals of distinct integers, a subset of the harmonic series. They aren't unique for a given real number, but the *greedy* Egyptian fraction for that real number is unique. Has anyone inspected the greedy Egyptian fraction for pi? The first few terms are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 744, 1173268, 2586625801171, 14348276635209672362238685, 1062286904072440687703470835520966381484062674280821 (A243020). See the pattern? Me neither. Except the first 12 terms. Maybe if it continued that way? After all, a friend of mine once confidently asserted that pi was infinite. (I think he may have been exaggerating.) Speaking of infinite, try to find an Egyptian fraction for a thousand. Good luck. Unlike the Egyptian fraction for pi, it's finite. But it contains more terms than there are atoms in the known universe, most of them consecutive integers starting with 1. What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original.