The sum of a prime power and an almost prime
Year: 2025
Type: article
Abstract: For any fixed k ≥ 2 {k\geq 2} , we prove that every sufficiently large integer can be expressed as the sum of a kth power of a prime and a number with at most M ( k ) = 6 k {M(k)=6k} prime factors. For sufficiently large k we also show that one can take M ( k ) = ( 2 + ε ) k {M(k)=(2+\varepsilon)k} for any ε > 0 {\varepsilon>0} , or M ( k ) = ( 1 + ε ) k {M(k)=(1+\varepsilon)k} under the assumption of the Elliott–Halberstam conjecture. Moreover, we give a variant of this result which accounts for congruence conditions and strengthens a classical theorem of Erdős and Rao. The main tools we employ are the weighted sieve method of Diamond, Halberstam and Richert, bounds on the number of representations of an integer as the sum of two kth powers, and results on kth power residues. We also use some simple computations and arguments to conjecture an optimal value of M ( k ) {M(k)} , as well as a related variant of Hardy and Littlewood’s Conjecture H.
Source: Forum Mathematicum
Authors: Daniel R. Johnston, Simon N. Thomas
Institutions: UNSW Canberra, The University of Queensland
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,712
OpenAlex ID: W4410880522