On Fri, 19 Oct 2012 01:06:32 +0100, Peter Percival wrote:
> What squares of positive integers are the sums of two or more distinct > cubes of positive integers? > > 9 = 1 + 8, 36 = 1 + 8 + 27, 100 = 1 + 8 + 27 + 64, 225 = 1 + 8 + 27 + 64 + > 125, etc. > > On the LHSs are the squares of the triangular numbers. Are there any > examples not in that sequence?
As noted in <http://mathworld.wolfram.com/CubicNumber.html>, every positive number is a sum of no more than 9 positive cubes, "proved by Dickson, Pillai, and Niven in the early twentieth century", and every sufficiently large integer is a sum of no more than 7 positive cubes. "Deshouillers et al. (2000) conjectured that 7373170279850 is the largest integer that cannot be expressed as the sum of four nonnegative cubes".
Let S be the set of sums of three cubes of numbers that are in the range 0 to n. There are n^3 such sums. Even though many of the sums have duplicated values, S nevertheless covers on the order of n^3 values in the range from 0 to 3 n^3, so naively the chance of a number being the sum of three cubes is in the neighborhood of 1/3. Let T be the set of sums of four cubes of numbers that are in the range 0 to n. This gives some fraction of n^4 distinct sums for numbers in the range 0 to 4 n^3, so on the average there are in the neighborhood of n/4 ways to write a number less than 4 n^3 as the sum of 4 cubes. However, among even small squares it looks like 14^2, 18^2, 20^2, and 26^2 require five cubes in their sums: