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

  • 2
    $\begingroup$ your solution isn't relatively prime $\endgroup$ – J. W. Tanner Jul 4 at 1:02
  • 5
    $\begingroup$ $$16=0^3+2^3+2^3$$ 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
  • 4
    $\begingroup$ 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

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.

  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 2
    $\begingroup$ @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

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:


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

And for $c=1$ we get,


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.