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NOTE ON LEHMER–PIERCE SEQUENCES WITH THE SAME PRIME DIVISORS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 04 October 2017, pp. 11-14
- Print publication:
- February 2018
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- Article
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Let
$a_{1},a_{2},\ldots ,a_{m}$ and 
$b_{1},b_{2},\ldots ,b_{l}$ be two sequences of pairwise distinct positive integers greater than 
$1$. Assume also that none of the above numbers is a perfect power. If for each positive integer 
$n$ and prime number 
$p$ the number 
$\prod _{i=1}^{m}(1-a_{i}^{n})$ is divisible by 
$p$ if and only if the number 
$\prod _{j=1}^{l}(1-b_{j}^{n})$ is divisible by 
$p$, then 
$m=l$ and 
$\{a_{1},a_{2},\ldots ,a_{m}\}=\{b_{1},b_{2},\ldots ,b_{l}\}$.
