Gaps in semigroups

Abstract

In this paper we investigate the behaviour of the gaps in numerical semigroups. We will give an explicit formula for the i th gap of a semigroup generated by k + 1 consecutive integers (generalizing a result due to Brauer) as well as for a special numerical semigroup of three generators. It is also proved that the number of gaps of the numerical semigroup generated by integers p and q with g . c . d . ( p , q ) = 1 , in the interval [ pq - ( k + 1 ) ( p + q ) , … , pq - k ( p + q ) ] is equals to 2 ( k + 1 ) + kq p + kp q for each k = 1 , … , pq p + q - 1 . We actually give two proofs of the latter result, the first one uses the so-called Apery sets and the second one is an application of the well-known Pick's theorem.