Let 0≤λ be a real and p a prime number, with [p,p+λp] containing at least two primes. Denote by f_λ the largest integer which cannot be written as a sum of primes from [p,p+λp].
Then
f_λ∼⌊2+2/λ⌋p
as p goes to infnity.
Further a question of Wilf about the 'Money Changing Problem' has a positive answer for all semigroups of multiplicity p containing the primes from [p,2p].
In particular, this holds for the semigroup generated by all primes not less than p.
This repository contains the source code for the article On the Frobenius number of certain numerical semigroups by R. Waldi, M. Hellus and A. Rechenauer.
There are five little programs, implemented in GAP and Rust which show the numerical evidence for various assertions and conjectures in the article.
For large numbers the Rust versions are more performant than the
GAP implementations (these were mainly used for checking the results).
For example the computation of the table
t4_quotient_not_always_6
took about 24 hours on a modern desktop, while it seemed impossible
to wait for the results of the GAP version.
cargo b --release && ./target/release/t4
creates a csv-file use to check f/p for primes in T(4)
cargo b --release && ./target/release/all_lambda
creates a csv-file which varies q while
computing the ng generated by the primes in the
interval [p..q] for the fixed prime 48623
cargo b --release && ./target/release/full
creates a csv which computes the numerical semigroup generated by all primes not less then p for various primes p.
cargo b --release && ./target/release/plots >../target/plots_from_rust.csv
creates a csv which is used by the python script py/plot.py to draw some pictures
cargo b --release && ./target/release/frobeniusnumber
creates a csv which computes the numerical semigroup generated by all primes between p and 2p (and gives evidence for f/p ~ 4)
(this is the rust main.rs)