Lucas Numbers with Lehmer Property
Bernadette Faye, Florian Luca

TL;DR
This paper proves that no Lehmer numbers exist within the Lucas sequence, extending previous results that showed their absence in the Fibonacci sequence, using adapted mathematical methods.
Contribution
It demonstrates the non-existence of Lehmer numbers in the Lucas sequence, building on prior work on Fibonacci numbers with a novel adaptation of existing methods.
Findings
No Lehmer number in Lucas sequence
Extension of non-existence results from Fibonacci to Lucas
Method adaptation for Lucas sequence
Abstract
A composite positive integer n is Lehmer if \phi(n) divides n-1, where \phi(n) is the Euler's totient function. No Lehmer number is known, nor has it been proved that they don't exist. In 2007, the second author [7] proved that there is no Lehmer number in the Fibonacci sequence. In this paper, we adapt the method from [7] to show that there is no Lehmer number in the companion Lucas sequence of the Fibonacci sequence given by and for all
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Advanced Mathematical Identities · Analytic Number Theory Research
