Thanks, Rich! On Mon, May 23, 2016 at 8:45 PM, <rcs@xmission.com> wrote:
Base 10 is useful for mentally splitting a small number into two squares.
After removing factors of 2 and 5, the remaining cofactor of the target will end in digit 1,3,7,or9. For 1 and 9, the split must either be xxx00 + y^2, or xxe25 + y^2. (e is an even digit.) For 3 and 7, double the target to end in 6 or 4; the split must be xxe25 + y^2.
example: 2089 = a^2 + b^2, find all splits. If the split is xxx00+y^2, y^2 must end in 89. y must end in 17,33,67,or 89. 17 and 33 are in range. trying these, a^2 is 1800 or 1000. no joy. If the split is xxe25+y^2, y^2 must end in e64. Y must end in 8 or 92. (42 & 58 squared end in o64.) 8 is in range. 2089-64 = 2025, bingo. the only split. since gcd(8,45) is 1, 2089 is prime.
example: 1997: split 3994 into xxe25 + xxo69. y must end in 13,37,63,or 87. 87 is too big; the others give 3994-y^2 = 3825, 2625, 25. 3825 is excluded because the hundreds digit must be 0,2,or 6; 8 is not. 2625 is excluded because 26 is not of the form n^2 + n. (The bracketing squares for 26 are 25 and 36, and n2+n is near the middle.) 3994-63^2 = 25 scores. since 63 and 5 are relatively prime, 1997 is prime. The split is average & semidifference of 5,63: 34, 29; 1156+841.
Base ten is especially useful for this exercise, because 5 is a very small 4k+1 prime, and using an even base simplifies determining the residue of a number mod 4 or 8. (Odd squares must be 8k+1, explaining the e in e25.) Determining whether a number is odd or even, in an odd base, requires examining every digit.
---
re spigot: the spigot algorithm for e is "represent in factorial base, convert to decimal by repeatedly multiplying by 10 and printing the integer digit (and discarding it)". This works for any conventional base. Gosper's continued fraction arithmetic allows base conversion, and can be done as a spigot algorithm in a natural way. 10 is not special.
re BBP (not spigot):
I don't know of anything for pi except base = powers of 2.
For natural logs, You can fiddle around with logs of 1 + delta, where delta is "nice" in your chosen base. For base 3, this gets log of 2/3, 4/3, 10/9, 26/27, ... Some adds & subtracts will get you log of 2, 3, 5, 7, 11, 13, and some sporadic larger primes in ternary. Base 2 is even better, but I can't see how to separate out log23 from 2047 = 23*89. In base 10, you can get log of 9/10, 11/10, ..., of limited interest.
I think the BBP pi formula can be interpreted as a linear combination of logs with (small) complex deltas, but Mr. P is better qualified to enlighten us on this point.
Rich
--------------- Quoting Mike Stay <metaweta@gmail.com>:
Is there anything interesting at all number-theoretically about base 10? Binary is obviously special for a lot of reasons. Ternary has the smallest radix economy, being closest to e, but that's all I can think of off the top of my head for why it would be better than some other base. Poking through my list archive, Gosper's 2005 email "squarefree string draws Sierpinski triangle" talks about some interesting stuff you can do with the alternating sum of nonzero digits in balanced ternary. The BBP algorithm for pi uses base 16, but I don't understand why well enough to know if that's fundamental or whether there are spigot algorithms for pi in any base. There are probably special things about a base being prime. But is there anything interesting about base 10?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com