What's enigmatic about this kind of very simple expression for a near-integer is whether it is a coincidence, or whether it begs for an explanation. To explain or not to explain: that is the question. For example, it's fairly well known that exp(pi*sqrt(163)) is an integer up to < 10^(-12), and this can be explained by the q-expansion of the j-invariant. This explanation also holds for the near-integers exp(pi*sqrt(67), up to < 10^(-5), exp(pi*sqrt(43), up to < 10^(-3), exp(pi*sqrt(19), up to < 1/4. Wikipedia also mentions this amazing discovery of D. Bailey & J. Borwein: Integral_{x=0 to oo} (cos(2x) * Prod_{n=1 to oo} cos(x/n)) dx = pi/8, up to < 10^(-42). Wikipedia claims this is just a coincidence, but I'd call that chutzpah. --Dan Simon wrote: << exp(Pi)-Pi = 19,9990999... . . . found in the 80's by me, Neil and a certain John H. C.
________________________________________________________________________________________ It goes without saying that .
Here's an amusing article by the Borwein's about this and many other examples: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P56.pdf Victor On Fri, Apr 27, 2012 at 9:20 AM, Dan Asimov <dasimov@earthlink.net> wrote:
What's enigmatic about this kind of very simple expression for a near-integer is whether it is a coincidence, or whether it begs for an explanation.
To explain or not to explain: that is the question.
For example, it's fairly well known that
exp(pi*sqrt(163)) is an integer up to < 10^(-12), and this can be explained by the q-expansion of the j-invariant.
This explanation also holds for the near-integers exp(pi*sqrt(67), up to < 10^(-5), exp(pi*sqrt(43), up to < 10^(-3), exp(pi*sqrt(19), up to < 1/4.
Wikipedia also mentions this amazing discovery of D. Bailey & J. Borwein:
Integral_{x=0 to oo} (cos(2x) * Prod_{n=1 to oo} cos(x/n)) dx = pi/8, up to < 10^(-42).
Wikipedia claims this is just a coincidence, but I'd call that chutzpah.
--Dan
Simon wrote:
<< exp(Pi)-Pi = 19,9990999...
. . .
found in the 80's by me, Neil and a certain John H. C.
________________________________________________________________________________________ It goes without saying that .
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