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

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October 16

Combinatorial aspects in identities involving sum of n cubes divisibility

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)

One form of the general relationship that captures any number of variables is $p_{3}=e_{1}\cdot (p_{2}-e_{2})+3e_{3}$, where p is the power sum symmetric polynomial and e is the elementary symmetric polynomial. --JBL (talk) 02:00, 17 October 2019 (UTC)

Before anyone asks, higher powers are more complicated, but some analogous identities do exist. For example

$a^{4}+b^{4}+c^{4}=(a+b+c)(a^{3}+b^{3}+c^{3}-a^{2}b-a^{2}c-ab^{2}-ac^{2}-b^{2}c-bc^{2})+2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})$

$a^{5}+b^{5}+c^{5}=(a+b+c)({\rm {some\,factor}})-5(a^{2}b^{2}c+a^{2}bc^{2}+ab^{2}c^{2})$

$a^{6}+b^{6}+c^{6}=(a+b+c)({\rm {some\,factor}})-2(a^{3}b^{3}+a^{3}c^{3}+b^{3}c^{3})+9a^{2}b^{2}c^{2}$

$a^{7}+b^{7}+c^{7}=(a+b+c)({\rm {some\,factor}})+7(a^{3}b^{3}c+a^{3}bc^{3}+ab^{3}c^{3})$

$a^{8}+b^{8}+c^{8}=(a+b+c)({\rm {some\,factor}})+2(a^{4}b^{4}+a^{4}c^{4}+b^{4}c^{4})-16(a^{3}b^{3}c^{2}+a^{3}b^{2}c^{3}+a^{2}b^{3}c^{3})$

$a^{9}+b^{9}+c^{9}=(a+b+c)({\rm {some\,factor}})-9(a^{4}b^{4}c+a^{4}bc^{4}+ab^{4}c^{4})+30a^{3}b^{3}c^{3}$

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)