Jawaban:

5 poin

Jawab:Given k, a natural number,

determine if there exists x,y,z INTEGERS such that x3+y3+z3=k.

It is not obvious that this problem is

decidable (I think it is but have not been able to find an exact statement to that affect; however, if it was not solvable, I would know that, hence it is solvable. If you know a ref give it in the comments.)

If k= 4,5 mod 9 then mod arguments easily show there is no solution. Huisman showed that if ks 1000, k=1,2,3,6,7,8 mod 9 and max(lxl,ly1.Izl) < 1015 and k is NOT one of

33, 42, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975

then there was a solution. For those on the list it was unknown.

Recently Booker (not Cory Booker, the candidate for prez, but Andrew Booker who I assume is a math-computer science person and is not running for

prez) showed that

x3 + y3 + z3 =33

DOES have a solution in INTEGERS. It is

x=8,866,128,975,287,528

y=-8,778,405,442,862,239

z=-2,736,111,468,807,040

does that make us more likely or less

likely to think that

x3 + y3 + z3 =42

has a solution? How about 114, etc, the others on the list?

Rather than say what I think is true (I have no idea) here is what I HOPE is true: that the resolution of these problems leads to some mathematics of interest.

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