On an equation involving fractional powers with prime numbers of a special type

TL;DR

This paper proves the existence of solutions to a prime-based equation involving fractional powers within a specific range of c, where the primes satisfy certain factorization constraints on p+2.

Contribution

It establishes the solvability of a prime equation with fractional powers and factorization conditions, extending previous results to a new range of c values.

Findings

Solutions exist for sufficiently large N when 1 < c < 17/16.

Primes p_i satisfy that p_i + 2 has bounded prime factors.

The result applies to primes with specific factorization properties.

Abstract

We consider the equation $[p_{1}^{c}] + [p_{2}^{c}] + [p_{3}^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{17}{16}$, then it has a solution in prime numbers $p_{1}$, $p_{2}$, $p_{3}$ such that each of the numbers $p_{1} + 2$, $p_{2} + 2$, $p_{3} + 2$ has at most $\left [ \frac{95}{17 - 16c} \right ]$ prime factors, counted with the multiplicity.