Rich writes: << I can't put my hand on the book, but I think Vaughn has shown that every sufficiently large number is of the form a^2+b^3+c^5. The key is that the sum 1/2 + 1/3 + 1/5 = 31/30 > 1, so the expected number of representations for a number N is, on average, K * N^1/30.
and << This is false when the sum of 1/a + 1/b + 1/c < 1. The set of available sums < N is less than the product of the possible x,y,z values, N^(1/a) * N(1/b) * N^(1/c), so the average spacing is at least N ^ ( 1 - 1/a - 1/b - 1/c). The interesting cases are abc = 22* 233 234 235 236 244 333.
A mere coincidence perhaps, but 1/a + 1/b + 1/c = R(a,b,c) > 1 <=> a sphere of Gaussian curvature K == 1 can be tiled by geodesic triangles whose angles are pi/a, pi/b, pi/c. (Just as R(a,b,c)=1 <=> the case of the plane with K==0, and R(a,b,c) < 1 <=> the case of the (hyperbolic) plane with K == -1). --Dan