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# Two Questions watch

1. One: are there any triples of integers (a,b,c) other than the two obvious ones such that a+b+c=sqrt(a^3+b^3+c^3)? (The two obvious ones being (0,0,0) and (1,2,3))

Two: Can anyone give me the primitive roots of unity for z^5 in radical form? I couldn't be arsed to algebraically solve the resulting biquadratic.
2. (Original post by J.F.N)
Two: Can anyone give me the primitive roots of unity for z^5 in radical form? I couldn't be arsed to algebraically solve the resulting biquadratic.
Hint: they solve z^4 + z^3 + z^2 + z + 1 = 0

so

(z^2 + 1/z^2) + (z + 1/z) + 1 = 0

Set y = z + 1/z and you get

(y^2 - 2) + y + 1 = 0

3. (Original post by J.F.N)
One: are there any triples of integers (a,b,c) other than the two obvious ones such that a+b+c=sqrt(a^3+b^3+c^3)? (The two obvious ones being (0,0,0) and (1,2,3))
(3,3,3) is surely another obvious one

Will think on the others later.
4. (1, 0, 0) and its 2 rearrangements are also solutions.

I tried proving that these solutions are the only ones by using modulo 7 considerations, but I kept going round in circles. Is there an algebraic solution to this problem?
5. (Original post by dvs)
(1, 0, 0) and its 2 rearrangements are also solutions.

I tried proving that these solutions are the only ones by using modulo 7 considerations, but I kept going round in circles. Is there an algebraic solution to this problem?
But 1,0,0 is not the only solution (its rearrangements are trivial in any case). Thinking about the problem since, these are all the solutions I know of so far:

(0,0,0),(a,-a,0),(a,-a,1),(0,1,2),(1,2,3),(3,3,3)

Out of those, perhaps only (1,2,3) and (3,3,3) are 'interesting'; i've also shown that no more triples than the ones listed above can contain a 1. Anyway, there must be a way to solve it--the literature on Diophantine equations is so vast that such an equation must have been considered before.
6. I meant that I tried to find all solutions or prove that no other ones exist (other than the ones we found).

I did some google searching and I think this problem is tied somehow to the (unsolved?) one that searches for all numbers expressible as a sum of three cubes.
7. (Original post by dvs)
I did some google searching and I think this problem is tied somehow to the (unsolved?) one that searches for all numbers expressible as a sum of three cubes.
I ran a program that shows the density of numbers expressible as the sum of three cubes; it seems that there are infinitely many such numbers. It is certainly not true that every number is expressible as such, as hundreds of exceptions are known. Actually finding expressions for those which you can write as a sum of three cubes is of course extremely difficult.

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Updated: July 12, 2005
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