6

Given $x$ and $y$ as sums of nine cubes. When we multiply them, $xy$ may again be written as a sum of nine cubes, due to Wieferich and/or Kempner.

Does this invent a 9 dimensional number system, just like the complex number forms a 2 dimensional number system?1

If number system is not the right terminus, what do we have here?

$${ }$$ $${ }$$

1 inspired by comments to this answer

Community
  • 1
draks ...
  • 18,755
  • 3
    Fundamentally I believe these are different concepts. Complex numbers are formed by field extension. We add a symbol $i$ to our field ($\mathbf{R}$ in this case) which is a solution to the otherwise unsolvable equation $x^2+1=0$. The nine cube theorem, while interesting in its own regard, offers a sort of decomposition of natural numbers into cubes. Much like prime factorization gives us each natural number as a product of primes. – doppz Dec 07 '13 at 23:08
  • Mostly, the number systems I know about all depend on sums of squares somewhere along the line: $x^2+y^2$ for the norm of a complex number, and something similar for quaternions. Not sure if a sum of cubes would fit into this scheme - but an interesting question! – Old John Dec 07 '13 at 23:12
  • @doppz do you think that the (rare) unique decompositions of natural numbers in to cubes or higher powers play a role here? – draks ... Dec 07 '13 at 23:25
  • 2
    An interesting question. IDK, but do Wieferich and Kemper have something like an identity of the form (sum of cubes of $a_i$) times (sum of cubes of $b_i$)= (sum of cubes of nine bilinear combos of $a_ib_j$)? The corresponding identity involving four squares comes from the reduced norm of Hamilton quaternions. There are 9-dimensional division algebras - see this example by Matt E for such an algebra over rationals and this example by me for an algebra over $\Bbb{Q}[\sqrt{-3}]$. – Jyrki Lahtonen Dec 08 '13 at 17:50
  • 1
    (cont'd) The reduced norm of those algebras is a homogeneous cubic form on the nine coordinates. The reduced norm is just the determinant of that matrix representing the algebra (see my answer for its structure). But the reduced norm won't be a sum of nine cubes. This is because the reduced norm does not vanish except when the nine coordinates are all zeros. May be another central simple algebra would have a reduced norm that is a sum of 9 cubes? IDK. – Jyrki Lahtonen Dec 08 '13 at 17:54

0 Answers0