The paper investigates integer solutions to Diophantine equations related to Lehmer's totient conjecture. The authors introduce the concept of "spoof factorizations", which are partial factorizations of integers where the bases and exponents are allowed to be any positive integers. They define the spoof evaluation and spoof totient functions, and study "spoof Lehmer factorizations" - those spoof factorizations that satisfy a condition analogous to Lehmer's totient conjecture.
The authors prove that for any integer r ≥ 2, the Diophantine equation defining spoof Lehmer factorizations has only finitely many integer solutions with bases at least 2, and these solutions can be explicitly computed. They provide an algorithm to find all nontrivial positive spoof Lehmer factorizations with a fixed number of bases.
Running the algorithm, the authors enumerate all nontrivial positive spoof Lehmer factorizations with 6 or fewer factors, and identify several infinite families of such factorizations. They also discuss potential extensions of their work to allow negative bases, and suggest avenues for future research.
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by Grant Molnar... lúc arxiv.org 09-26-2024
https://arxiv.org/pdf/2409.17076.pdfYêu cầu sâu hơn