Abstract
For two relatively prime square-free positive integers a and b, we study integers of the form \(a p + b P_2\), where ap and \(b P_2\) are both square-free, p is a prime, and \(P_2\) has at most two prime divisors. If a and b are both odd, we prove that every sufficiently large even integer relatively prime to ab can be written in this form; if one of a and b is even, we prove every sufficiently large odd integer relatively prime to ab can be written in this form. This generalizes and improves Chen’s Theorem, Estermann’s theorem on sums of primes and square-free numbers, and results on Lemoine’s conjecture.
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The author’s research is partially supported by the Fundamental Research Funds for the Central Universities, Nankai University (Grant No. 63221040) and the National Natural Science Foundation of China (Grant No. 12201313).
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Li, H. Additive representations of natural numbers. Ramanujan J 60, 999–1024 (2023). https://doi.org/10.1007/s11139-022-00649-2
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DOI: https://doi.org/10.1007/s11139-022-00649-2