On a Diophantine equation involving fractional powers with primes of special types

Abstract

Suppose that $ N $ is a sufficiently large real number. In this paper it is proved that for $ 2 < c < \frac{990}{479} $, the Diophantine equation $ \left[p_{1}^{c}\right]+\left[p_{2}^{c}\right]+\left[p_{3}^{c}\right]+\left[p_{4}^{c}\right]+\left[p_{5}^{c}\right] = N $ is solvable in primes $ p_{1}, p_{2}, p_{3}, p_4, p_5 $ such that each of the numbers $ p_{i}+2, i = 1, 2, 3, 4, 5 $ has at most $ \left[\frac{6227}{3960-1916c}\right] $ prime factors.

Keywords

• almost prime
• diophantine equality
• fractional powers
• exponential sum