Equal sums of three powers
Let \(N_d(B)\) denote the number of positive integer solutions of the equation \[ x_1^d+x_2^d+x_3^d=x_4^d+x_5^d+x_6^d, \] with \(\max x_i\leq B\). The latter equation is satisfied, of course, when \(x_4\), \(x_5\), \(x_6\) are a permutation of \(x_1\), \(x_2\), \(x_3\). The trivial solutions of this type contribute \(6B^3+O(B^2)\) to \(N_d(B)\), and write \(N_d^{(0)}(B)\) for the number of non-trivial solutions counted by \(N_d(B)\). Then the following noteworthy estimate is established in this paper: for any \(d\geq25\) and any \(\varepsilon>0\), one has \[ N_d^{(0)}(B)\ll B^{3-1/24+\varepsilon}+ B^{5/2+5/(2\sqrt d)+11/(6(d-1))+\varepsilon}+ B^{41/16+5/(2\sqrt d)+\varepsilon}. \] Thus, in particular, \(N_d^{(0)}(B)=o(B^3)\) for \(d\geq33\). The authors point out that there would be little trouble to prove the same bound in the case where we count the integer solutions \(x_i\), not necessarily positive, subject to \(\max| x_i| \leq B\), with altering the meaning of ``trivial'' suitably. They also mention that the methods of this paper could be employed to treat arbitrary diagonal equations of the form \(a_1x^d+\cdots+a_6x_6^d=0\), with rational coefficients \(a_i\). There are various alternative ways to express the principal theorem above. And for example, it is derived as a corollary that on putting \(c=\Gamma(1+1/d)^3/\Gamma(1+3/d)\), for \(d\geq33\), there are asymptotically \((c/6)x^{3/d}\) natural numbers \(n\leq x\) which are sums of three \(d\)th powers, and almost all of these have essentially just one representation.
scientific article; zbMATH DE number 2104979