Proof of the theorem on numerical semigroups

Question

$S$ - numerical semigroup, generated by $a$, $b$.

$S=(a,b)={s0 < s1 < s2 < .....}$

$c$ - conductor of numerical semigroup: $c=(a-1)(b-1)$.

All elements $x_i >= c$ are incremented by one.

How can I prove it?

My teacher wrote in (see pictures). Which famous formula did he use?

Tell me, what is the formula. Next I will try to prove the theorem itself.

What is the formula you want the name of? – AnotherPerson Oct 20 '16 at 02:22 — AnotherPerson, Oct 20 '16 at 02:22
https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity ?? – Oct 20 '16 at 02:54 — , Oct 20 '16 at 02:54
Byron Schmuland, yes!!! Thank you!!!!! – Maria Oct 20 '16 at 03:24 — Maria, Oct 20 '16 at 03:24

score 2 · Answer 1 · answered Oct 20 '16 at 02:52

I think you're referring to the ua+vb=1 part? This is an idea I've often heard referred to as "the Euclidean algorithm". Technically this is an algorithm for finding appropriate coefficients, but it's correctness of course also implies the theorem that such coefficients are guaranteed to exist (whenever a and b are coprime).

Proof of the theorem on numerical semigroups

1 Answers1