In 1947, R. P. Bambah and S. Chowla proved that if ? > 2 p2 then, for all large integers n, there are integers u, v such that n ? u2 + v2 < n + ?n 1 4 . We pose a hypothesis on the distribution of the fractional parts of numbers involving the square root function, and show that, on the assumption of the hypothesis, the same result holds for any ? > 0. The case n ? u2 + v2 < n + 2 p2n 1 4 a, with a 0, is also considered.
展开▼
