Bill Gosper <billgosper@gmail.com> wrote:
Digits. Feh. What a waste of chips and neurons. The only excuse: Now that they've done it, converting to the continued fraction might be slightly easier than extracting the cf directly.
Why decimal digits? Why pi? Why not 3^3^3 balanced ternary digits of the cube root of 3? Some people like continued fractions. More people like decimal digits. Some people like e, phi, or Euler's constant. More people like pi. Pi was proven irrational in the 18th century, and transcendental in the 19th. The 20th belonged to digit hunters. I hope the 21st will find something new to do with the number, such as a proof (or disproof) of normality. (Tests of normality are a common excuse for digit hunters, as if they needed one.) If trends continue, a mole of decimal digits of pi will be known before the end of the century. Perhaps we could all agree to stop there. So we're a third of the way to the next term of this series: 1400 1706 1949 1958 1961 1973 1983 1987 1989 1997 1999 2002 2011... (The nth term is when pi was first known to 10^n digits.) Plouffe's algorithm that gives the nth binary digit of pi without giving any others may lead to a proof of pi's normality in binary, something that previously appeared completely opaque in every base. At the very least, it's much easier to store, or even memorize, the quadrillionth binary digit of pi than to store, email, or memorize trillions of decimal digits. There is, however, a shortcut: Here are all the decimal digits of pi: 0123456789. (Some assembly (and repetition) required.) Instead of digit hunting a constant, I've been searching for an order-five additive-multiplicative magic square. This has the advantage of a definite end point. (Unless, of course, there isn't one.) So far, my search has resembled digit hunting. I've found more than a thousand semi-magic squares (rows and columns work, but diagonals do not). Once I have enough such squares, I plan to do statistical tests on near-misses to see where fully magic squares might be lurking, or at least where I can find lots more semi-magic squares. For instance is there any pattern to the common sums or the common products? What about their parity? For the products, there's a clear pattern there: All but one of the more than a thousand squares I've found has an even product. For the sums, odd is more likely than even by a statisticly significant amount, but not enough that you'd likely notice by eyeballing a list of the sums.