Year: 2020
Type: article
Abstract: Let $$p_1=2, p_2=3, p_3=5, \ldots$$ be the consecutive prime numbers, $$S_n$$ the numerical semigroup generated by the primes not less than $$p_n$$ and $$u_n$$ the largest irredundant generator of $$S_n$$ . We will show, that $$u_n\sim 3p_n$$ . Similarly, for the largest integer $$f_n$$ not contained in $$S_n$$ , by computational evidence ( https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Hellus/table_1.pdf ) we suspect that (1) $$f_n$$ is an odd number for $$n\ge 5$$ and (2) $$f_n\sim 3p_n$$ ; further (3) $$4p_n>f_{n+1}$$ for $$n\ge 1$$ . If $$f_n$$ is odd for large n, then $$f_n\sim 3p_n$$ . In case $$f_n\sim 3p_n$$ every large even integer x is the sum of two primes. If $$4p_n>f_{n+1}$$ for $$n\ge 1$$ , then the Goldbach conjecture holds true. Further, Wilf’s question in Wilf (Am Math Mon 85:562–565, 1978) has a positive answer for the semigroups $$S_n$$ .
Source: Semigroup Forum
Authors: Michael Hellus, Anton Rechenauer, Rolf Waldi
Institution: University of Regensburg
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,750
OpenAlex ID: W2969923786