We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.

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We show that if h ∈ Z[x] is a polynomial of degree k ≥ 2 such that h(N) contains a multiple of q for every q ∈ N, known as an intersective polynomial, then any subset of {1, 2, . . . , N } with no nonzero differences of the form h(n) for n ∈ N has density at most a constant depending on h and c times (log N ) −c log log log log N , for any c < (log((k 2 + k)/2)) −1 . Bounds of this type were previously known only for monomials and intersective quadratics, and this is currently the best-known bound for the original Furstenberg-Sárközy Theorem, i.e. h(n) = n 2 . The intersective condition is necessary to force any density decay for polynomial difference-free sets, and in that sense our result is the maximal extension of this particular quantitative estimate. Further, we show that if g, h ∈ Z[x] are intersective, then any set lacking nonzero differences of the form g(m) + h(n) for m, n ∈ N has density at most exp(−c(log N ) µ ), where c = c(g, h) > 0, µ = µ(deg(g), deg(h)) > 0, and µ(2, 2) = 1/2. We also include a brief discussion of sums of three or more polynomials in the final section.2000 Mathematics Subject Classification. 11B30.

We extend the best known bound on the largest subset of {1, 2, . . . , N } with no square differences to the largest possible class of quadratic polynomials. 2000 Mathematics Subject Classification. 11B30. 1 A set A ⊆ N is said to have positive upper density if lim sup N→∞ |A∩[1,N]| N > 0, where [1, N ] denotes {1, 2, . . . , N }.

We define a necessary and sufficient condition on a polynomial h ∈ Z[x] to guarantee that every set of natural numbers of positive upper density contains a nonzero difference of the form h(p) for some prime p. Moreover, we establish a quantitative estimate on the size of the largest subset of {1, 2, . . . , N } which lacks the desired arithmetic structure, showing that if deg(h) = k, then the density of such a set is at most a constant times (log N ) −c for any c < 1/(2k − 2). We also discuss how an improved version of this result for k = 2 and a relative version in the primes can be obtained with some additional known methods.2000 Mathematics Subject Classification. 11B30.