论文标题
将Pillai的问题扩展到高斯线
Extending a problem of Pillai to Gaussian lines
论文作者
论文摘要
令$ l $为原始的高斯线，即，在复杂平面中包含两个的线路，因此无限的codrime高斯整数。我们证明存在一个整数$ g_l $，因此，对于每个整数$ n \ geq g_l $，在$ l $上连续$ n $的$ n $ l $ coprime中都没有一个属性的$ n $连续的高斯整数序列无限的序列。我们还调查了最小的整数$ g_l $，以便$ l $包含$ g_l $的序列，并带有此属性的连续高斯整数。我们表明$ g_l \ neq g_l $一般。另外，每条高斯线$ l $ $ g_l \ geq 7 $，我们为$ g_l = 7 $提供必要的条件，并用$ g_l \ geq 260,000 $描述许多高斯线。我们推测$ g_l $和$ g_l $都可以任意大。我们的结果将Pillai的众所周知问题从理性整数到高斯整数。
Let $L$ be a primitive Gaussian line, that is, a line in the complex plane that contains two, and hence infinitely many, coprime Gaussian integers. We prove that there exists an integer $G_L$ such that for every integer $n\geq G_L$ there are infinitely many sequences of $n$ consecutive Gaussian integers on $L$ with the property that none of the Gaussian integers in the sequence is coprime to all the others. We also investigate the smallest integer $g_L$ such that $L$ contains a sequence of $g_L$ consecutive Gaussian integers with this property. We show that $g_L\neq G_L$ in general. Also, $g_L\geq 7$ for every Gaussian line $L$, and we give necessary and sufficient conditions for $g_L=7$ and describe infinitely many Gaussian lines with $g_L\geq 260,000$. We conjecture that both $g_L$ and $G_L$ can be arbitrarily large. Our results extend a well-known problem of Pillai from the rational integers to the Gaussian integers.
