The reverse Goldbach problem and a refined Zsiflaw--Legeis theorem
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910243118645248 |
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| author | Harm, Michael Johnston, Daniel R. |
| author_facet | Harm, Michael Johnston, Daniel R. |
| contents | We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at representations of integers as the sum of primes and reversed primes. We show that:
(1) Every large odd integer is the sum of a prime and two reversed primes ($N=p_1+\overleftarrow{p_2}+\overleftarrow{p_3}$).
(2) Every large odd integer is the sum of two primes and a reversed prime ($N=p_1+p_2+\overleftarrow{p_3}$).
(3) Almost all even integers are the sum of a prime and a reversed prime ($N=p_1+\overleftarrow{p_2}$).
(4) All large integers are the sum of a reversed prime and a square-free number ($N=\overleftarrow{p}+η$, $μ^2(η)=1$).
To obtain our results, along with associated asymptotics, we apply the Hardy--Littlewood circle method and a novel refinement of the ``Zsiflaw--Legeis" theorem on the distribution of reversed primes in arithmetic progressions. Notably, our variant of the Zsiflaw--Legeis theorem does not require one to fix the digit length unlike previous versions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21876 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The reverse Goldbach problem and a refined Zsiflaw--Legeis theorem Harm, Michael Johnston, Daniel R. Number Theory 11A63, 11P32 (Primary) 11L07, 11N13, 11P55 (Secondary) We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at representations of integers as the sum of primes and reversed primes. We show that: (1) Every large odd integer is the sum of a prime and two reversed primes ($N=p_1+\overleftarrow{p_2}+\overleftarrow{p_3}$). (2) Every large odd integer is the sum of two primes and a reversed prime ($N=p_1+p_2+\overleftarrow{p_3}$). (3) Almost all even integers are the sum of a prime and a reversed prime ($N=p_1+\overleftarrow{p_2}$). (4) All large integers are the sum of a reversed prime and a square-free number ($N=\overleftarrow{p}+η$, $μ^2(η)=1$). To obtain our results, along with associated asymptotics, we apply the Hardy--Littlewood circle method and a novel refinement of the ``Zsiflaw--Legeis" theorem on the distribution of reversed primes in arithmetic progressions. Notably, our variant of the Zsiflaw--Legeis theorem does not require one to fix the digit length unlike previous versions. |
| title | The reverse Goldbach problem and a refined Zsiflaw--Legeis theorem |
| topic | Number Theory 11A63, 11P32 (Primary) 11L07, 11N13, 11P55 (Secondary) |
| url | https://arxiv.org/abs/2605.21876 |