On 9/29/13, Warren D Smith <warren.wds@gmail.com> wrote:

...

Hum, interesting but what is t ? what type of number is it ? what is the rule for t ? Simon Plouffe

--well, I'd like to know that myself... I do not have a full explanation/understanding for what is going on here... However, I observe the following pattern for the particular t that I gave:

t = 101000100000000111111111111111100000000000000000000000000000001(binary)

note the "gap sizes" between the 1's (or -1's) are 1, 3, 7, 15, 31 a sequence I'm sure you recognize.

--This sort of pattern happens for b=2^31 and b=2^63, but for b=2^f for other f, it is not as easy to see what "the pattern" is. For b=2^31, here are t such that ContinuedFraction(t/b) has only 1s and 2's, and t is a sum of a small number of signed powers of 2: decimal binary 5653987329=101010001000000010000000000000001 5620432895=101001111000000001111111111111111 5117116417=100110001000000010000000000000001 5083561983=100101111000000001111111111111111 5117116415=100110001000000001111111111111111 5653987327=101010001000000001111111111111111 Here are some random examples for b=2^f for other f: b=2^61: t=110101110010010011111001001110000000000000000000000000000001 b=2^61: t=1001010001101101100000110110001111111111111111111111111111111 b=2^59: t=1101100000010111111111111111000000000000000000000000000001 b=2^59: t=10011110101010111111111111110111111111111111111111111111111 b=2^51: t=11011000010111111111111100000000000000000000000001 b=2^51: t=11011000010111111111111011111111111111111111111111 b=2^45: t=100111101100110010111100000000000000000000001