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I'm messing around with the Sum of Three Cubes problem, and I'm seeing something interesting:

For the case $n=16$, all the sources I've seen[1][2][3] are saying that the smallest solution is

$$16 = (-511)^3 + (-1609)^3 + (1626)^3$$

...But using my own very naive algorithm, I was able to find the following solution:

$$16 = (-48)^3 + (-94)^3 + (98)^3$$

Am I misunderstanding the problem in some way? Or is this something that was simply overlooked by the authors of the sources? Surely it's not a novel finding worthy of publication?

[1] http://oeis.org/A060465

[2] http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100.htm

[3] https://en.wikipedia.org/wiki/Sums_of_three_cubes

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    $\begingroup$ your solution isn't relatively prime $\endgroup$ – J. W. Tanner Jul 4 at 1:02
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    $\begingroup$ $$16=0^3+2^3+2^3$$ is trivial if you don't want relatively prime. $\endgroup$ – N. S. Jul 4 at 1:05
  • $\begingroup$ $-1609$ is the smallest of the six numbers you mention. $\endgroup$ – Piquito Jul 4 at 1:24
  • $\begingroup$ @Piquito No, $-1609$ is the lowest of the six numbers. The smallest is $-48$. $\endgroup$ – Erick Wong Jul 4 at 1:33
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    $\begingroup$ You are correct that it’s not a novel finding since the solution to $n=2$ quickly generates this solution (or as N.S. observes, an even smaller one). But you have correctly discovered that the OEIS title does not accurately describe the sequence. $\endgroup$ – Erick Wong Jul 4 at 1:36
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The numbers you cubed are all divisible by $2$, which divides $16$,

but according to the OEIS link, only primitive solutions of $x^3+y^3+z^3=n,$

where $\gcd(x,y,z)$ does not divide $n,$ are considered.

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    $\begingroup$ That explains the discrepancy, but the OEIS title is simply erroneous since it’s inconsistent with this restriction. Both versions of “smallest” are worthy of consideration, and neither of the non-OEIS sources seem to care about primitivity at all. It seems to be an extension of the conjecture to show that a primitive solution even exists. $\endgroup$ – Erick Wong Jul 4 at 1:10
  • $\begingroup$ @ErickWong From OEIS,"COMMENTS Indexed by A060464. Only primitive solutions where gcd(x,y,z) does not divide n are considered. " $\endgroup$ – N. S. Jul 4 at 1:17
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    $\begingroup$ Aha, that explains it. In that case I agree with @ErickWong that the other sources should clarify this point! $\endgroup$ – Dmitry Brant Jul 4 at 1:18
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    $\begingroup$ @N.S. As I said, the title of the OEIS entry does not mention primitivity and therefore specifies a precise object which is NOT the one in the comments, nor does it match the sequence values. The title is wrong and should be corrected. $\endgroup$ – Erick Wong Jul 4 at 1:29
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Solution given by "OP",

$16 =(98)^3 +(-48)^3 + (-94)^3$ ----$(1)$

When $(1)$ is divided by $'8'$ we get,

$2=(49)^3+(-24)^3+(-47)^3$ ----$(2)$

On the internet there is parametric solution:

$2=(1+6c^3)^3+(1-6c^3)^3+(-6c^2)^3$

For, $c=2$ we get solution by "OP"

And for $c=1$ we get,

$2=7^3-6^3-5^3$

Which is even smaller than the one given by "OP".

So essentially solution given by "OP"[equation $(2)$]

is not the smallest. But "OP" could upload his "new"

Algoritm and maybe it could be use-full in the search

for other integer 'n'for which solution's are not

known for $(n=a^3+b^3+c^3)$.

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