We prove that the Cohen-Macaulay type of an almost Gorenstein monomial curve C ⊆ A 4 is at most 3, and make some considerations on the general case.

Given coprime positive integers a 1 < · · · < a d , the Frobenius number F is the largest integer which is not representable as a non-negative integer combination of the a i . Let g denote the number of all non-representable positive integers: Wilf conjectured that d ≥ F+1 F+1−g . We prove that for every fixed value of a 1 d the conjecture holds for all values of a 1 which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality d = F+1 F+1−g .

We present a fast algorithm to compute the Delta set of a nonsymmetric numerical semigroup with embedding dimension three. We also characterize the sets of integers that are the Delta set of a numerical semigroup of this kind.

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these resutls, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.2010 Mathematics Subject Classification. 05A17,20M13,20M14.

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set {n 2 , (n+1) 2 , . . .} is asymptotically O(n 2 ), verifying thus a conjecture of Dutch and Rickett. Furthermore we ask a question on the representation of integers as sum of four large squares.2010 Mathematics Subject Classification. 11D07,11P05.