In 1939, Erd\H{o}s and Mahler conjectured that an irrational real number $\xi$ must be a Liouville number whenever $P(p_nq_n)$ is bounded for infinitely many convergents $p_n/q_n$, where $P(N)$ denotes the largest prime factor of a nonzero integer $N$. In this note, we prove a neighbouring-denominator case of their conjecture: if $P(p_nq_nq_{n+1})$ is bounded for infinitely many $n$, then $\xi$ is a Liouville number. The proof combines the determinant identity for consecutive convergents with a fixed-base estimate for linear forms in $p$-adic logarithms.
In 1943, Erd\H{o}s considered the minimum number $f(n)$ of terms between two fractions in the Farey sequence of order $n$ whose numerators and denominators are oppositely ordered. Determining the constant $c$ in $f(n)=(c+o(1))n$ is known as Erd\H{o}s Problem 1005. Recently, Cipollini solved this asymptotic problem by p...
Let $P^+(n)$ denote the largest prime factor of $n$. One of Erd\H{o}s and Tur\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is 1/2. In this paper, we prove that this density is larger than 0.280, which improves the previous result 0.2017 by L\"u and Wang (2025). We a...
Let $a(n)$ be the sequence A028342 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\sum_{n\ge0} a(n)x^n/n! = \prod_{i\ge1}(1-x^i)^{-1/i}$. Equivalently, $a(n)$ counts permutations of an $n$-element labeled set in which every cycle is assigned one divisor of its l...
We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first $N$ orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we p...
Fix a prime power $q$. Let $D_q(n)$ be the largest possible least degree of a polynomial omitted by a non-covering family of $n$ congruence classes in $\mathbb F_q[x]$. Assuming the known theorem that every non-covering family of $n$ classes omits a polynomial of degree less than $n$, we prove \[ D_q(n)=\frac{n}{q-1}+O...
Cameron and Erd\H{o}s asked if the number of sets free of arithmetic progressions of length $k$ is $2^{r_k(n)(1+o(1))}$, where $r_k(n)$ is the maximum cardinality of a $k$-AP-free subset of $\{1, \dots, n\}$. Balogh, Liu and Sharifzadeh made significant progress on this question showing that it is $2^{O(r_k(n))}$ for a...
Patrick Morris, Miquel Ortega, Juanjo Ru'e· 0 citations
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