I find this beautiful...

Mathguy15

It would be interesting if one could prove that 1 is never one of the integers in the solution.

EDIT:Oh, and thank you micromass for that link. I will definitely look into it.

micromass

It will be since

[tex]1999=1^3+321409^3+894590^3+(-908211)^3[/tex]

Mathguy15

Wow, how did you get that? Did you use some kind of computer or did you find it using some theory or theorem?

micromass

Just letting a computer check the possibilities. Nothing fancy.
I didn't even program it, I found this on a list online

Anti-Crackpot

Hi Mathguy15,

Multiply out the terms, but first cube your 3rd term since you're going to cube it later anyway...

n^3 + 30*n^2 + 300n + 1000
-n^3 + 30*n^2 - 300n + 1000
(-60*n^2)
- 1

→

30*n^2 + 30*n^2 - 60*n^2 = 0
1000 + 1000 - 1 = 1999

You could do the same thing for (n+8)^3 = 1n^3 + 24*n^2 + 192n + 512, and (n-8)^3, but you'd have to replace (-60*n^2) with (-48*n^2) and 1999 with 1023. And so on...

Note that you're not really dealing with cubes since they cancel out when you multiply out and collect the terms.

alphachapmtl

Seems you're right!

http://i41.tinypic.com/107pcvr.png

Anti-Crackpot

If you want to generalize further mathguy15...

for any n or k in N

First term: (1*k^0*n^3) + (3*k^1*n^2) + (3*k^2*n^1) + (1*k^3*n^0) = (n+k)^3
Second term: -(1*k^0*n^3) + (3*k^1*n^2) - (3*k^2*n^1) + (1*k^3*n^0)
Third term: (Derived from above 2nd terms doubled)
Fourth term: -1

... then figure out the formula for those coefficients (x) above (which is almost certainly related to the Binomial Coefficients aka "the coefficients of the expansion of (x+1)^n"). Do that and then you'll have a nice little equation in 3 variables that will work for any n, k or x in N.

Binomial Coefficients:
1
1 1 --> Powers of 1
1 2 1 --> Powers of 2
1 3 3 1 --> Powers of 3...
1 4 6 4 1 --> Powers of 4...

...when inserted into Polynomials.

In the example you posted, then set k = 10

- AC

Anti-Crackpot

Here's an example of what I was referring to in the previous post, thinking-wise. Take the form...

(10 + n)^5 + (10 - n)^5 + (-100*n^4) + (-20000*n^2) + (-1) = 199999

Echoing the form used by mathguy15...

v = (10 + n)
w = (10 - n)
x = (-100*n^4)^(1/5)
y = (-20000*n'^2))^(1/5)
z = -1

x = an integer for (-100*(100*s^5)^4)^(1/5); n =(100*s^5)
y = an integer for (-20000*(125*s'^10)^2))^(1/5); n' = (125*s^10)

Will have to see where to go from there, meaning what conclusions one might come to regarding:
v^5 + w^5 + x^5 + y^5 + z^5