论文标题
广场弗罗贝尼乌斯编号
The Square Frobenius Number
论文作者
论文摘要
令$ s = \ left \ langle s_1，\ ldots，s_n \ right \ rangle $是由相对主要的正整数生成的数值半群，$ s_1，\ ldots，s_n $。令$ k \ geqslant 2 $为整数。 In this paper, we consider the following $k$-power variant of the Frobenius number of $S$ defined as $${}^{k\!}r\!\left(S\right):= \text{ the largest } k \text{-power integer not belonging to } S.$$In this paper, we investigate the case $k=2$.我们为$ {}^{2 \！} r \！\ left（s_a \ right）$提供一个上限，用于{\ em arithmetic processions}生成的无限族semogroups $ s_a $。后者事实证明是$ {}^{2 \！} r \！\ left的确切值（\ left \ langle s_1，s_2 \ right \ rangle \ right \ right）$在某些条件下。我们提出了$ {}^{2 \！} r \！\ left（\ left \ langle s_1，s_1+d \ right \ rangle \ right）$时的确切公式\ right \ rangle \ right）$和$ {}^{2 \！} r \！\ left（\ left \ langle s_1，s_1+2 \ 2 \ right \ rangle \ right \ right）$，并提出两个相关的猜想。我们终于讨论了一些相关的问题。
Let $S=\left\langle s_1,\ldots,s_n\right\rangle$ be a numerical semigroup generated by the relatively prime positive integers $s_1,\ldots,s_n$. Let $k\geqslant 2$ be an integer. In this paper, we consider the following $k$-power variant of the Frobenius number of $S$ defined as $${}^{k\!}r\!\left(S\right):= \text{ the largest } k \text{-power integer not belonging to } S.$$In this paper, we investigate the case $k=2$. We give an upper bound for ${}^{2\!}r\!\left(S_A\right)$ for an infinite family of semigroups $S_A$ generated by {\em arithmetic progressions}. The latter turns out to be the exact value of ${}^{2\!}r\!\left(\left\langle s_1,s_2\right\rangle\right)$ under certain conditions. We present an exact formula for ${}^{2\!}r\!\left(\left\langle s_1,s_1+d \right\rangle\right)$ when $d=3,4$ and $5$, study ${}^{2\!}r\!\left(\left\langle s_1,s_1+1 \right\rangle\right)$ and ${}^{2\!}r\!\left(\left\langle s_1,s_1+2 \right\rangle\right)$ and put forward two relevant conjectures. We finally discuss some related questions.
