Wednesday, 23 September 2009

An Interesting Observation, and a Problem

I came across a couple of neat things reading old journals sent to me by Dave Renfro... here are two that seem worth sharing:
In the Mathematical Spectrum in 1984, Malcolm smithers of the Open University shared a discovery... "I found the following interesting number on my Oric 1 48K" (The Oric1 (16k) was released in 1982 and was the first color computer to sale for under 100 GB Pounds..I still remember working with my son as we both learned computing on a Tandy color computer... alas he has left me in the dust computer wise)

"3435 = $3^{3} + 4^{4} +3^{3} + 5^{5}$" which left me wondering, is there any number between this number and 1=$1^{1}$ that replicates this kind of behavior.. none came quickly to mind, and I was too busy to do a computer search.

Here is an interesting problem from the American Mathematical Monthly from May, 1930:

A piece of pie is cut from a pie with a radius r. The cut is a sector with a central angle of $\theta$. Assuming that the angle $\theta$ is < 180o. What is the radius of the smallest plate that will hold the cut piece of pie?
Have fun. For the calculus student, is the function differentiable over this domain?
Enjoy

Joshua Zucker said...

Other than 1, and possibly 0 depending on what you think 0^0 is equal to, there aren't many more. There's a not-too-hard proof that there can only be finitely many. A computer search up into the multi-millions doesn't find any more besides 3435.

(I assumed 0^0 = 1 for all my work)

Joshua Zucker said...

By the way, http://www.research.att.com/~njas/sequences/A046253 will spoil your fun if you are interested in working further on this problem.

Pat's Blog said...

Well I leave out 0^0 as a case (I just call it undefined) but I think it is remarkable that there is actually another, and that it is SOOOOOO very large... 438,579,088l... so there probably are more...but gosh, how big must they be... apparently even the on-line inter sequence didn't find the next one..

Joshua Zucker said...

No, there aren't any more. At the end of the OEIS entry you can see the keyword "full" which means that all the terms are listed here (and "fini" which means there are finitely many terms).

Joshua Zucker said...

And by the way, you're wrong. You do say 0^0 = 1, not undefined. Unless when you say "The binomial theorem states ..." you include "unless x+y = 0 and n = 0, in which case ..."

Same with people who claim they define trapezoid as a quadrilateral having exactly one pair of parallel sides. They all still say the "trapezoid rule" for integrals, not the "trapezoid or rectangle", and they find the area of those trapezoids, not distinguishing two cases depending on whether f(x0) and f(x1) are the same ...

There are other places where theorems would have annoying exceptions if 0^0 were anything but 1. I have yet to find any meaningful theorem where I'd be wrong to define it as 1.

I was even thinking that something trivial like 0^x = 0 might be such a theorem, but it's not. I say "0^x = 0 for all x > 0" and the people who think 0^0 is undefined say exactly the same thing.