Problems

Does there exist A=a_1<a_2<…⊂ N which is a minimal basis of order 2 (every large integer is the sum of 2 elements from A, and no proper subset of A has this property) such that lim_k→ ∞a_k/k^2=c for some c≠ 0? A claimed construction gives…

If A(x) counts integers satisfying the Sylow divisor condition, determine the constant c in A(x)/x = exp(-(c + o(1)) √log x loglog x). The claimed exact value is c = 1/(2√log 2).

Let A=1≤ a_1< a_2<… be a set of integers such that Abackslash B is complete for any finite subset B and not complete for any infinite subset B. If a_n+1/a_n ≥ 1+ε for all n, must lim_n a_n+1/a_n=(1+√5)/2? Under the reading where the ratio…

How large must y(ε, n) be so that every interval (x, x+y) contains at most ε y integers having a divisor in (n, 2n)? The candidate proof gives the sharp fixed-ε order y = Θ_ε(n), uniformly in the translate.

If n_1 < n_2 < … with n_k+1/n_k ≥ c > 1, must ∑_k 1/F_n_k be irrational? The proposed proof closes the range 1 < c < 2 left open by earlier criteria.

If A is a forbidden-divisor set with |A ∩ [1,x]| = o(√x) and B = b_1 < b_2 < … the sifted set, must x^-1 ∑_b_i < x (b_i+1 - b_i)^2 converge to a finite limit?

If h(r) is the maximal finite exact order attainable by an additive basis of order at most r, what is lim_r → ∞ h(r)/r^2? The candidate proof identifies the sharp limit 1/3.

Estimate the number F(x) of minimal distinct covering systems whose moduli all lie in [1, x]. The candidate proof gives loglog F(x)/log x → 1, i.e. F(x) = exp(x^1+o(1)).

If g_3(n) is the largest size of A ⊆ [1,n] with fewer than three representations of every product a_1 a_2, does its conjectured second-order normalized term converge? The candidate proof gives an explicit limit constant.

If each integer has at most r representations m = pa with p prime and a ∈ A ⊆ [1, N], what is the best upper bound for ∑_a ∈ A 1/a? The candidate proof gives the matching order Θ_r(log N / loglog N).

Let F(N) be the maximal size of A⊆1,…,N such that no a∈ A divides the sum of any nonempty subset of A∖a. Estimate F(N). The lower bound F(N)≫ N^1/5 is classical, from constructions of Erdős and Csaba, and every non-dividing set is…

If a/b ∈ Q_>0 and b is squarefree, can a/b always be written as a finite sum of reciprocals of distinct products of two distinct primes?

Let a_1 = 2 and a_2 = 3 and continue the sequence by appending to a_1, …, a_n all possible values of a_ia_j - 1 with i ≠ j. Is it true that the set of integers which eventually appear has positive density?

If A ⊆ N has unbounded dyadic-shell counts and ∑_n ∈ A |θ n| = ∞ for every 0 < θ < 1, must A be complete - is every sufficiently large integer a sum of distinct elements of A?

For the least t_k(n) with n | t_k(n)(t_k(n)+1)…(t_k(n)+k-1), do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with c = 1/2048 admissible in the t_2 bound.