Jump to content

Wikipedia:Reference desk/Archives/Mathematics/2019 October 16

From Wikipedia, the free encyclopedia
Mathematics desk
< October 15 << Sep | October | Nov >> October 17 >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.


October 16

[edit]

Combinatorial aspects in identities involving sum of n cubes divisibility

[edit]

Starting from the previous cases of identities involving sum of cubes of positive integers (3 and 4 cubes) divided by the sum of numbers present in cubes, what is the general formula for the sum of n cubes divided by the sum of n numbers? What is the general form of sum of products of succesive cases of the form 3(abc + abd + bcd + acd) (4 choose 3 or 4C3 or C(4,3)) encountered in the case of four cubes compared with sum of 5 cubes, 6 cubes, and so on. (C(n,3)?)..?--109.166.137.3 (talk) 18:36, 16 October 2019 (UTC)[reply]

One form of the general relationship that captures any number of variables is , where p is the power sum symmetric polynomial and e is the elementary symmetric polynomial. --JBL (talk) 02:00, 17 October 2019 (UTC)[reply]
Before anyone asks, higher powers are more complicated, but some analogous identities do exist. For example
with the formulas getting longer and longer. With more variables yet more complicated identities exist; I'm not going to work them out though. --RDBury (talk) 09:14, 17 October 2019 (UTC)[reply]