Math curiosity

[Lepton]

a^2 + b^2 = c^2
3^2 + 4^2 = 5^2
.......Everyone knows Pythagores' Theorem.

a^3 + b^3 + c^3 = d^3
3^3 + 4^3 + 5^3 = 6^3
........Fancy, eh?

What does d^3 signify?
(I don't know the answer myself)

[Chuck]

Mere coincidence. It means nothing.

[Bicho the Inhaler]

"There are no coincidences in the natural numbers." or something to that effect, said by someone or other famous.

[daniel801]

it means that three ice cubes of sides 3, 4, and 5 when melted would fit in an ice cube tray with side 6

[daniel801]

...if re-frozen

[Lepton]

But no funky trigangle-y goodness?

[groza528]

Not that I know of (displays title)

[Suspence]

I found the quote to which Mr. Inhaler refers:

"Achilles: An "infinite coincidence"? Among the natural numbers, NOTHING is coincidental-nothing happens without there being some underlying pattern. "

It has to mean something somewhere...but what???

------------------
I hate people who try to write interesting things in their signature

[Bicho the Inhaler]

Yes, page 398 of Hoftstadter's GEB. That must have been it.

[Coyote]

http://mathforum.org/dr.math/problems/toscano7.14.97.html

I don't really understand why Fermat's Last Theorem being proven implies that no formula for finding solutions to a^3+b^3+c^3=d^3 exists, but if it's true, I doubt there'll be any 'practical' application similar to the Pythagorean Theory.

The article lists several other solutions to the equation though, just in case anyone wants to look them over and try to figure out some kind of connecting idea to them.

[dave10000]

For no good reason, I wrote the following Mathematica program:

While[
(a = Random[Integer, {1000, 1500}]) ||
(b = Random[Integer, {1000, 1500}]) ||
(c = Random[Integer, {1000, 1500}]) ||
!IntegerQ[(a^3 + b^3 + c^3)^(1/3)]]
Print[a]
Print[b]
Print[c]
Print[(a^3 + b^3 + c^3)^(1/3)]

In under 2 minutes of run time, my computer told me that:

1004^3 + 1219^3 + 1332^3 = 1731^3