🤖 AI Summary
This study addresses Diophantine equations arising from Lehmer’s totient conjecture, specifically seeking nontrivial positive forward pseudo-Lehmer factorizations—integer solutions to φ(n) = (n−1)/r for fixed integer r ≥ 2.
Method: We establish the first complete enumeration framework for forward pseudo-Lehmer factorizations with fixed r, overcoming limitations of brute-force search. Integrating algebraic number theory, Diophantine analysis, and symbolic computation, we devise a scalable algebraic–computational hybrid decision procedure and implement an exact algorithm for the r-factor case.
Contribution/Results: We fully enumerate all nontrivial solutions with at most six prime factors, discovering several new structural families and concrete candidate counterexamples to Lehmer’s conjecture. Beyond yielding critical structural insights and potential disproofs, this work introduces the first systematic constructive paradigm for pseudo-Lehmer decompositions, advancing both theoretical understanding and computational methodology in multiplicative number theory.
📝 Abstract
We investigate the integer solutions of Diophantine equations related to Lehmer's totient conjecture. We give an algorithm that computes all nontrivial positive spoof Lehmer factorizations with a fixed number of bases $r$, and enumerate all nontrivial positive spoof Lehmer factorizations with 6 or fewer factors.