Let $p$ be an odd prime and let $A,B\subseteq\mathbb{F}_p$ be nonempty and distinct, with $|A|\ge |B|$. We give a complete classification of the pairs for which the mixed restricted sumset $A\mathbin{\widehat+} B=\{x+y:x\in A,\ y\in B,\ x\ne y\}$ has size $\min\{p,|A|+|B|-2\}$. The main new contribution concerns the in...
We study the reducibility over $\mathbb{Q}$ of $F_{p,e}(x)=rx^m+p^ef(x)$, where $r\in\mathbb{Z}\setminus\{0\}$, $f\in\mathbb{Z}[x]$, and $0\le m<n:=\mathrm{deg}\, f$, for primes $p$ above explicit coefficient-dependent thresholds. For arbitrary $e\ge1$, we determine the degrees, endpoint $p$-adic valuations, reductions...
We determine the complete ordered solution set of the Fermat-type matrix equation $X^n+Y^n=Z^n$ in $\mathrm{GL}_2(\mathbb{Z})$ for every integer $n\ge3$, expressed in terms of simultaneous integral-conjugacy orbits while keeping the variable symmetries of the equation separate. The equation is solvable if and only if $...
Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved...
Hong-Jian Li, Ping-Zhi Yuan, Shijie Yuan et al.· 0 citations
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