# Sum of three cubes for $n=16$

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?

• your solution isn't relatively prime – J. W. Tanner Jul 4 '19 at 1:02
• $$16=0^3+2^3+2^3$$ is trivial if you don't want relatively prime. – N. S. Jul 4 '19 at 1:05
• $-1609$ is the smallest of the six numbers you mention. – Piquito Jul 4 '19 at 1:24
• @Piquito No, $-1609$ is the lowest of the six numbers. The smallest is $-48$. – Erick Wong Jul 4 '19 at 1:33
• 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. – Erick Wong Jul 4 '19 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.

• 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. – Erick Wong Jul 4 '19 at 1:10
• @ErickWong From OEIS,"COMMENTS Indexed by A060464. Only primitive solutions where gcd(x,y,z) does not divide n are considered. " – N. S. Jul 4 '19 at 1:17
• Aha, that explains it. In that case I agree with @ErickWong that the other sources should clarify this point! – Dmitry Brant Jul 4 '19 at 1:18
• @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. – Erick Wong Jul 4 '19 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:

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

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