On the number of representations of integers as the sum of a prime and a $k$-th power.
Aran Nayebi
posted in Mathematics on Thursday, October 15th, 2009
Let $R_k(n)$ be the number of representations of an integer $n$ as the sum of a prime and a $k$-th power for $k \ge 2$. Furthermore, set $E_k(X) = |\{n \le X, n \ne m^k, n\text{not a sum of a prime and a $k$-th power}\}|$. </p> <p>In the present paper we use sieve techniques to obtain a strong upper-bound on $R_k(n)$ for $n \le X$ with no exceptions, and we improve upon the results of A. Zaccagnini to prove $E_k(X) \ll_{k} X^{0.9819}$. We also briefly outline methods that can significantly improve the latter result to $E_k(X) \ll_{k} X^{0.7}$.
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