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Lehmer's Totient Problem and a Prime-Cost Pivot Method | Synapse

February 2, 20260 citationsOpen Access

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Lehmer's Totient Problem and a Prime-Cost Pivot Method

DSDaniel Sautot

Key Points

This research aims to resolve Lehmer's totient problem related to composite integers and the phi function.
Proved that no composite integer n satisfies phi(n) | (n - 1)
Used a prime-cost pivot inequality to rule out large parameters
Developed a finite-range certificate that is machine-checkable
Verified results using deterministic scripts
Proved that phi(n) does not divide (n - 1) for any composite integer n
Provided code and associated artefacts for verification in a companion repository

Abstract

This record provides the preprint "Lehmer’s Totient Problem and a Prime-Cost Pivot Method". It resolves Lehmer’s totient problem by proving that no composite integer n satisfies phi(n) | (n − 1). The proof combines a prime-cost pivot inequality ruling out the large-parameter regime with a finite-range, machine-checkable certificate for the remaining window, verified by deterministic scripts. Companion materials (code and certificate artefacts) are provided in the associated companion repository for this record.

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Cite This Study

Daniel Sautot (Sun,) studied this question.

synapsesocial.com/papers/6980ffc6c1c9540dea8128ff https://doi.org/https://doi.org/10.5281/zenodo.18447108