Positive spoof Lehmer factorizations
Grant Molnar, Guntas Singh
TL;DR
This work generalizes Lehmer's totient conjecture by replacing prime bases with spoof bases in a factorization framework, defining spoof totient $\widetilde{\varphi}$ and spoof evaluation $\widetilde{\epsilon}$ and studying the Diophantine equation $k \cdot \widetilde{\varphi}(F) = \widetilde{\epsilon}(F) - 1$. It proves finiteness of nontrivial solutions for any fixed number of bases $r$ and develops a constructive algorithm to enumerate all such spoof Lehmer factorizations, supported by the $L$ and $U$ bounds and the $\kappa_r$ function. The paper identifies infinite families of spoof factorizations and catalogs 31 odd and 45 even solutions with at most six factors, providing code and data for reproducibility. These results illustrate the arithmetic richness of spoof analogues and underscore that any proof of Lehmer's conjecture would need primality-specific constraints beyond the algebraic structure alone. The work thereby blends theoretical bounds with computational enumeration to map the landscape of spoof Lehmer factorizations up to six factors.
Abstract
We investigate the integer solutions of Diophantine equations related to Lehmer's totient conjecture. We give an algorithm that computes all nontrivial spoof Lehmer factorizations with a fixed number of factors, and enumerate all nontrivial spoof Lehmer factorizations with 6 or fewer factors.