Equations Work!

With some help from this post, my blog now supports awesome looking equations (no more blurry bitmaps!). As such, I no longer hesitate to note that E=m c^2 , e^{\pi i}+1=0, and of course that \frac{d}{dx}\int_C^x f(t) dt = f(x) .
Cheers!
Ninjinuity

Note: Equations seem to look best in Firefox. They don't work at all in IE (no surprises there), and they don't quite work in Chrome. I'll keep looking into this. The script also seems to be picking up weird elements of the page and replacing them. I need to stop that.

Edit: The old equations code was inconsistent and had a habit of changing stuff not inside it's tags. With this new back end, they should work everywhere.

Math Puzzler #3

There are six Dudeney numbers, positive integers whose decimal digit sum cubed is equal to the original number. they are
amath 1=1^3=(1)^3 endmath,
amath 512=8^3=(5+1+2)^3 endmath,
amath 4913=17^3=(4+9+1+3)^3 endmath,
amath 5832=18^3=(5+8+3+2)^3 endmath,
amath 17576=26^3=(1+7+5+7+6)^3 endmath,
and amath 19683=27^3=(1+9+6+8+3)^3 amath.

1. Prove there are no larger Dudeney numbers.
2. Find all numbers where the fourth power of the digit sum is equal to the number itself.

Hint: A computer may be useful to check cases, but it is possible to do this problem by hand, albeit with a bit of paper.
Solution 1 after the jump, Solution 2 next week.

Still pinching myself.

I waited for a day to make this post, to make sure there was no mistake.

I got in to MIT. I suppose its hard to communicate how excited I am about this. It's been a major goal since middle school, but always kind of the ideal that I shot for, never expecting to really make it. Now I'm in. Surreal feeling, let me tell you.