## Sums of integer cubes

##### Abstract

Taxicab numbers, of Hardy and Ramanujan fame, are positive integers which can be represented as the sum of two positive integer cubes, in two distinct ways. The smallest such integer is 1729 = 13 + 123 = 93 + 103. One question which naturally arises, is to ask how many numbers with this property there are, up to some bound N. This is usually denoted (N). The current best lower bound, (N) > CN1/3 log(N), is due to Hooley. The best upper bound, (N) = O N4/9+ , is due to Heath-Brown. A related question is to count the number of integer solutions to w3+x3+y3+z3 = 0. A solution is considered trivial if it is some permutation of the form w3+(?w)3+y3+(?y)3 = 0. Manin’s conjecture states that the number of non-trivial solutions, with |w|, |x|, |y|, |z| < N1/3 should be asymptotic to cN1/3(log(N))4 for some positive constant c. Using a parametrization found by Euler, we show that the number of such solutions is in fact bounded below as predicted by Manin’s conjecture. Moreover, we show that by restricting ourselves to the case where two of w, x, y, z are non-negative and the other two are non-positive (that is, a solution which yields a taxicab number) we get the same lower bound cN1/3(log(N))4, though not necessarily the same constant.