论文标题
整数序列的在线百科全书的序列A348480的长解决方案
Long Solutions of Sequence A348480 of the On-Line Encyclopedia of Integer Sequences
论文作者
论文摘要
对于数字$ x $ coprime至$ 10 $，存在无限的许多二进制数字$ b $，因此最大的$ b $和rev（$ b $）= $ x $的最大共同除数，$ b = x $的数字总和（rev（$ b $）是$ b $的数字逆转）。在大多数情况下，满足这两个约束的最小$ b $仅包含一些零。但是在某些情况下，例如$ x = 7、11、13 $和$ 37 $，$ b $必须包含比零更大的零，而这些$ b $称为长解决方案。从这些是多孔数字的事实中，它直接以$ 11 $和37美元的价格。本文以$ 7 $和$ 13的价格，证明它们具有长期解决方案的证明。
For numbers $x$ coprime to $10$ there exist infinitely many binary numbers $b$ such that the greatest common divisor of $b$ and rev($b$) = $x$ and the sum of digits of $b = x$ (rev($b$) is the digit reversal of $b$). In most cases, the smallest $b$ that fulfill these two constraints contain just a few zeros. But in some cases like for $x = 7, 11, 13$ and $37$, $b$ must contain more zeros than ones and these $b$ are called long solutions. For $11$ and $37$ it follows directly from the fact that these are porous numbers. For $7$ and $13$, the proofs that they have long solutions are presented in this paper.
