For cubic or lower degree polynomials, p(x, y) = c in general seems to have an unbounded number of solutions, e.g. there are c for which a^3 + b^3 = c has an arbitrarily large number of solutions, vis A011541.
For quartic or higher polynomials, this does not appear to be the case. For example, a^4 + b^4 = c seems to have at most two distinct solutions in a, b for any c, vis A018768. At the time I computed A018768, I computed it way beyond the published values, but found no values with more than two representations. To my knowledge, no one has ever proved that there cannot be three representations, but no one has ever found an example.
sum(a..b, i^3) = p(b) - p(a-1) where p(n) = (n*(n+1)/2)^2, a quartic. Thus I might expect sum(a..b, i^3) = c to have a bounded number of distinct solutions.
It's an interesting question, no? Anyone have any ideas? Then modify this page! Thanks. —User:GraemeMcRaetalk22:35, November 18, 2012 (UTC)