Representation of integers as sums of fractional powers of primes and powers of 2
Volume 181 / 2017
Acta Arithmetica 181 (2017), 185-196
MSC: 11P32, 11P05, 11N36.
DOI: 10.4064/aa8663-5-2017
Published online: 9 November 2017
Abstract
Let $c$ be a real number with $1 \lt c \lt 2$. We consider the representation of integers in the form $$N=[p_1^c]+[p_2^c]+2^{\nu_1}+\cdots+2^{\nu_k},$$ where $p$ and $\nu$ denote a prime number and a positive integer respectively. We prove that when $1 \lt c \lt 29/28$, there exists an integer $k$ depending on $c$ such that each large integer $N$ can be represented in the form above.
