Short intervals asymptotic formulae for binary problems with prime powers

Abstract

We prove results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_1^{{\ell }_1}+p_2^{{\ell }_2}$, with ${\ell }_1,{\ell }_2\in \{2,3\}$, ${\ell }_1+{\ell }_2\le 5$ are fixed integers, and $n=p^{{\ell }_1}+m^{{\ell }_2}$, with ${\ell }_1=2$ and $2\le {\ell }_2\le 11$ or ${\ell }_1=3$ and ${\ell }_2=2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.

arXiv commentAccepted (Nov. 2017) for publication in Journal de Theorie des Nombres de Bordeaux. Two references updated

References in corpus (4)

• Sum of one prime and two squares of primes in short intervals
• On a ternary Diophantine problem with mixed powers of primes
• Short intervals asymptotic formulae for binary problems with primes and powers, II: density $1$
• Short intervals asymptotic formulae for binary problems with primes and powers, I: density $3/2$

Cited by in corpus (1)

• Short intervals asymptotic formulae for binary problems with prime powers, II