On the Equation ax ≡ x (mod b)
On the Equation ax ≡ x (mod b)
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关于方程 ax == x (mod b)
DOI：
10.1515/integ.2009.050
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Jam Germain
中科院分区：
文献类型：
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作者：
Jam Germain
摘要最近Jiménez-Urroz和Yebra [On the Equation ax x（mod bn）]对任意给定的a和B，构造了标题方程的解x.此外，他们还展示了如何将这些提升到B的更高次幂，以获得某些整数B的b进解。本文求出了该方程的所有正整数解x，证明了对给定的a和B，存在X/B + O B（1）解x ≤ X.我们还展示了如何解决方案可能会解除更一般性。此外，我们表明，建设[Jiménez Urroz和Yebra，在方程ax x（mod bn）]（和明显的修改）不能总是找到所有的解决方案ax x（mod B）。
Abstract Recently Jiménez-Urroz and Yebra [On the Equation ax ≡ x (mod bn )] constructed, for any given a and b, solutions x to the title equation. Moreover they showed how these can be lifted to higher powers of b to obtain a b-adic solution for certain integers b. In this paper we find all positive integer solutions x to the title equation, proving that, for given a and b, there are X/b + Ob (1) solutions x ≤ X. We also show how solutions may be lifted in more generality. Moreover we show that the construction of [Jiménez-Urroz and Yebra, On the Equation ax ≡ x (mod bn )] (and obvious modifications) cannot always find all solutions to ax ≡ x (mod b).