Problems

For the Erdős–Pomerance functions F(n) and h_P(n) counting how many consecutive integers are needed to contain a distinct multiple of each integer, respectively prime, up to n, the paper proves F(n) ≥ h_P(n) ≥ nexp((log 2/2 - o(1))log…

22 conjectures of Cohen about cyclic numbers (integers with gcd(n, φ(n)) = 1) settled at once - 16 proved, 6 disproved - together with a complete resolution of a related OEIS problem on sequences whose running averages are Fibonacci…

Let n_k be the least integer greater than 2k for which ∏_i=1^k (n_k - i) has no prime factor in (k, 2k). How rapidly must n_k grow?

Erdos and Graham asked whether a positive-density subset of 1,…,N can avoid having any two distinct elements a,b whose unit fractions average to a unit fraction. It can: there is a constant c>0 such that for all large N some A ⊆ 1,…,N of…

Banks and Martin conjectured in 2013 that for a primitive set A and any set Q of primes, the Erdos sum of the members of A composed only of primes in Q is at most the corresponding sum over Q itself. The unrestricted form turned out to be…

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.

The Lonely Runner Conjecture of Wills and Cusick states that among k+1 runners at distinct constant speeds on a unit circle, each runner is at some time at distance at least 1/(k+1) from all others. Following Rosenfeld's computer-assisted…

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…

For every real ξ>0 the sequence of integer parts [ξ 7^n], n=0,1,2,…, contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~7.

Regev and Stephens-Davidowitz conjectured that Z^n maximizes the Gaussian mass Θ_L(t) = ∑_x ∈ L e^-t|x|^2 over stable lattices for every t > 0. The sharp inequality holds for every integral unimodular lattice of rank n ≤ 32, with equality…

Let x_n = tan(∑_k=1^n arctan k). Amdeberhan, Medina and Moll conjectured that x_n ∉ Z for every n ≥ 5. Any integer value x_n = m must satisfy |m| ≥ e^(1/2+o(1)) n log n, which forces #1 ≤ n ≤ N : x_n ∈ Z = O(log N). The conjecture…

The paper gives counterexamples in dimensions eight and nine to two problems: the Cartesian-product problem posed by Cassels for critical determinants and formulated by Zong for lattice packings, and a question raised by Sarnak, formulated…