论文标题
魔术矩形，签名的魔术阵列和整数$λ$ - 折亲量heffter阵列
Magic rectangles, signed magic arrays and integer $λ$-fold relative Heffter arrays
论文作者
论文摘要
令$ m，n，s，k $为整数，以便$ 4 \ leq s \ leq n $，$ 4 \ leq k \ leq m $和$ ms = nk $。令$λ$为$ 2MS $的除数，让$ t $为$ \ frac {2ms}λ$的除数。在本文中，我们构建魔术矩形$ MR（m，n; s，k）$，签名魔术阵列$ sma（m，n; s，k）$和整数$λ$ - 倍相对相对heffter阵列$ {}^}^λh_t（m，n; s，s，k）$ s，k $ s，k $甚至是整数。特别是，我们证明所有$ m，n; s，k）$均为所有$ m，n，s，k $满足以前的假设。此外，我们证明存在一个$ MR（M，N; S，K）$和一个Integer $ {}^λH_T（m，n; s，k）$在以下情况下：$（$（i）$ $ s，k \ equiv 0 \ equiv 0 \ pmod 4 $; $（ii）$ $ s \ equiv 2 \ pmod 4 $和$ k \ equiv 0 \ pmod 4 $; $（iii）$ $ s \ equiv 0 \ pmod 4 $和$ k \ equiv 2 \ pmod 4 $; $（iv）$ $ s，k \ equiv 2 \ pmod 4 $和$ m，n $均匀。
Let $m,n,s,k$ be integers such that $4\leq s\leq n$, $4\leq k \leq m$ and $ms=nk$. Let $λ$ be a divisor of $2ms$ and let $t$ be a divisor of $\frac{2ms}λ$. In this paper we construct magic rectangles $MR(m,n;s,k)$, signed magic arrays $SMA(m,n;s,k)$ and integer $λ$-fold relative Heffter arrays ${}^λH_t(m,n;s,k)$ where $s,k$ are even integers. In particular, we prove that there exists an $SMA(m,n;s,k)$ for all $m,n,s,k$ satisfying the previous hypotheses. Furthermore, we prove that there exist an $MR(m,n;s,k)$ and an integer ${}^λH_t(m,n;s,k)$ in each of the following cases: $(i)$ $s,k \equiv 0 \pmod 4$; $(ii)$ $s\equiv 2\pmod 4$ and $k\equiv 0 \pmod 4$; $(iii)$ $s\equiv 0\pmod 4$ and $k\equiv 2 \pmod 4$; $(iv)$ $s,k\equiv 2 \pmod 4$ and $m,n$ both even.
