Aside from n = 4 or 5 mod 9, which are all impossible, the next unresolved case is x^3 + y^3 + z^3 = 42. On Fri, Mar 8, 2019 at 7:06 PM Dan Asimov <dasimov@earthlink.net> wrote:

Very interesting, Alon Amit! (Can I please have some anonymized network data?)

Suppose f : Z^3 —> Z via

f(x,y,z) = x^3 + y^3 + z^3.

Then f can evidently take some very large triples to very small integers.

Question: ----- Given an integer N in Z:

What is the asymptotic behavior of the set

S_3(N) = ((x,y,z) in Z^3 | x^3 + y^3 + z^3 = N}

???

Meaning above all: What is the asymptotic behavior of the monotone function

#_3(N): Z+ —> Z>=0

where #_3(N)(t) denotes the number

card({(x,y,z) in Z^3 | x^3 + y^3 + z^3 = N —and— x^2 + y^2 + z^2 <= t}

And same for other polynomials.

—Dan

-----

James Buddenhagen writes: ----- 33=8866128975287528^3+(-8778405442862239)^3+(-2736111468807040)^3 found by Timothy Browning communicated on Quora by Alon Amit -----

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