Problems

Does there exist an integer polynomial f of degree at least two and a set A ⊆ Z such that every integer has a unique representation n = a + f(k)? A manuscript claims the thirteenth powers admit a tiling complement.

Let k≥ 3 and f_k(N) be the maximum of ∑_n∈ A1/n over all A⊆1,…,N containing no k subsets with the same pairwise least common multiple. Estimate f_k(N). The claimed answer: f_k(N)=(log N)^γ_k+o(1), where γ_k is a weighted generalization of…

For S(x) = #(a,b) : a + b ≤ x, σ(a) + σ(b) = σ(a+b), is S(x) ~ cx? The preprint claims S(x) grows faster than x (log x)^R for every fixed R, ruling out the linear asymptotic.

Let k≥ 3 and A be an additive basis of order k. Does there exist a constant c=c(k)>0 such that if r(n)≥ clog n for all large n (where r(n) counts representations of n as a sum of at most k elements of A) then A must contain a minimal basis…