On a Diophantine equation with prime variables

Abstract

<abstract> Let $ [\alpha] $ denote the integer part of the real number $ \alpha $, $ N $ be a sufficiently large integer and $ (\kappa, \lambda) $ be the exponent pair. In this paper, we prove that for $ 1 &lt; c &lt; \frac{3+3\kappa-\lambda}{3\kappa+2} $, the Diophantine equation $ [p_1^c]+[p_2^c]+[p_3^c] = N $ is solvable in prime variables $ p_1, p_2, p_3 $. If we take $ (\kappa, \lambda) = \left(\frac{81}{242}, \frac{132}{242}\right) $, we can get the range $ 1 &lt; c &lt; \frac{837}{727} $, which improves the previous result of Cai. </abstract>