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
