A Binary Additive Equation Involving Fractional Powers

with integers m1, m2; henceforth, [θ] denotes the integral part of θ. Subsequently, the range for c in this result was extended by Gritsenko [3] and Konyagin [5]. In particular, the latter author showed that (1) has solutions in integers m1, m2 for 1 < c < 3 2 and n sufficiently large. The analogous problem with prime variables is considerably more difficult, possibly at least as difficult as the binary Goldbach problem. The only progress in that direction is a result of Laporta [6], which states that if 1 < c < 17 16 , then almost all n (in the sense usually used in analytic number theory) can be represented in the form (1) with primes m1, m2. Recently, Balanzario, Garaev and Zuazua [1] considered the equation