On integers which are representable as sums of large squares

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set $\{n^2,(n+1)^2,\ldots \}$ 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.Comment: 6 pages. To appear in International Journal of Number Theor

The first elements of the quotient of a numerical semigroup by a positive integer

Given three pairwise coprime positive integers $a_1,a_2,a_3 \in \mathbb{Z}^+$ we show the existence of a relation between the sets of the first elements of the three quotients $\frac{\langle a_i,a_j \rangle}{a_k}$ that can be made for every $\{i.j,k\}=\{1,2,3\}$. Then we use this result to give an improved version of Johnson's semi-explicit formula for the Frobenius number $g(a_1,a_2,a_3)$ without restriction on the choice of $a_1,a_2,a_3$ and to give an explicit formula for a particular class of numerical semigroups.Comment: 7 page

On a conjecture of Wilf about the Frobenius number

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 $d \geq \frac{F+1}{F+1-g}$. We prove that for every fixed value of $\lceil \frac{a_1}{d} \rceil$ 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 $d = \frac{F+1}{F+1-g}$

Lacunary polynomials and compositions

This thesis deals with lacunary polynomial compositions, that is, polynomial compositions having a fixed number of terms, with an eye towards some arithmetic applications. More specifically, we start by giving some results on polynomial powers having few terms, and then show how these results can be applied to study integer perfect powers having few non-zero digits in their representation in a fixed basis. In relation to this last problem, we also show that, for any fixed basis, there are infinitely many perfect squares having a given number of non-zero digits in their representation, with very few exceptions (which have already been treated in the literature). We then proceed to study lacunary polynomial compositions, focusing on polynomial compositions having relatively many "pure" terms, and apply our results to the study of Universal Hilbert Sets generated by functions associated to linear recurrence relations having only simple roots. After that, we briefly discuss the general case, and provide some evidence towards a general question concerning the minimum number of terms of a composition, in function of the number of variables of the inner polynomial. In the last chapter, we shift our focus towards some additive problems related to the study of these compositions. In particular, we describe a problem concerning additive factorization lengths between terms appearing as exponents of our compositions; then, we study, in the context of additive monoids, an invariant codifying the variations between lengths of additive factorizations of the same integer, showing how these factorizations are very difficult to control even in the most simple cases