Problems

Let A(x) count n ≤ x such that every prime p | n has a divisor d > 1 of n with d ≡ 1 pmod p. Erdos asked whether A(x)/x = exp(-(c+o(1))√log xloglog x). It does, with c = 1/(2√log 2).

Let D_q(n) be the largest possible least degree of a polynomial omitted by a non-covering family of n distinct-modulus congruence classes in F_q[x]. What is its asymptotic size? The answer is D_q(n) = n/q-1 + O_q(1).

Let S(x) count ordered pairs (a,b) with a+b ≤ x and σ(a)+σ(b) = σ(a+b). Erdos asked whether S(x) ~ cx. The opposite extreme holds: for every R > 0, S(x)/(x(log x)^R) → ∞, so the count beats every fixed logarithmic scale.

Is it true that for every ε,η>0 there exists a k such that the density of n for which P(n(n+1)…(n+k))>n^1-ε is at least 1-η, where P(m) is the greatest prime divisor of m? A short argument via the Matomäki-Radziwiłł theorem establishes the…

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.

Ross introduced S-perfect numbers, integers expressible as 1 + ∑ λ_j d_j over their proper divisors with coefficients in S, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd…

Erdős and Graham asked whether binomnk with 1 ≤ k ≤ n/2 must always have a divisor ≤ n that is close to n, meaning bigger than a fixed constant times n. Settled in both directions: true when k is large enough as a function of n, but false…

Let A(n) be the least positive integer not dividing binom2nn. Erdos asked for the behaviour of A(n) for reasonable n. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals…

Zhi-Wei Sun conjectured a closed evaluation of a truncated Legendre-symbol determinant. For every prime p ≡ 3 pmod 4 it equals ⌊ (p-2)/3 ⌋^2 x, proved by reducing to inverse data for Chapman's full Legendre-symbol matrix and evaluating…

For an irreducible crystallographic root system of rank r with Coxeter number h, the paper proves that Au's normalized Witten zeta function has a simple pole at 2/h and evaluates its residue in closed form in terms of the Cartan…

For n ≥ 4, the natural scalar Poisson-summation certificates cannot prove the Regev-Stephens-Davidowitz Gaussian mass conjecture: any such certificate saturates, so the whole approach is blocked.

Does there exist a good pairwise-coprime sequence u_n with ∑ 1/u_n < ∞ and polynomial growth? What if one only requires u_n ≤ e^o(n)?

For an odd prime p, do Sun's normalized trigonometric permanents satisfy s_p < 0 ⇔ p ≡ 5 pmod12 and s'_p < 0 ⇔ p ≡ 7 pmod 8? Exact computation at p = 29 refutes both sign laws.

An n-divisor set contains a multiple of every integer from 1 to n. Umans and Wang proposed, as the arithmetic-progression form of their Strong (α,β)-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude,…

Let n_k be the least n > 2k such that (n-k)(n-k+1)…(n-1) has no prime factor in (k, 2k). Erdos conjectured a superpolynomial lower bound; for all large k, n_k > e^log^2 k / (20 loglog k).

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…

Every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes. This is the ω = 2 integer case of a problem of Erdos and Graham, left as a conjecture by Butler, Erdos and Graham, who proved the ω = 3…

Let M(n) be the supremum of ∑_a ∈ A 1/(n-a) over pairwise coprime A ⊂ [1,n). Erdos asked whether M(n) ≤ ∑_p<n 1/p + O(1) uniformly. The average order is settled: ∑_n ≤ N M(n) = e^-γ N loglog N + O(N).

Does the block 11 occur infinitely often in the base-2 expansion of the Erdős-Borwein constant E = ∑_n ≥ 1 1/2^n - 1? Posed by Crandall in 2012.

Define φ_k(n) = ∑_1 ≤ a ≤ n, (a,n)=1 a^k and D_s = k ≥ s : φ_s(n) | φ_k(n) for every n. Is D_1 = 1, 3, 15, as conjectured by Büyükaşik and collaborators?