Comment on "NMR Experiment Factors Numbers with Gauss Sums"

TL;DR

This paper critically examines NMR-based algorithms for factor checking, revealing they cannot directly determine factors without prior explicit factorization, thus challenging their claimed efficiency.

Contribution

It demonstrates that NMR schemes for factor checking inherently require prior knowledge of factors, questioning their effectiveness for direct factorization.

Findings

NMR schemes do not directly factor numbers.

Factor checking with NMR requires prior factor knowledge.

Challenges the claimed efficiency of NMR-based factor algorithms.

Abstract

Mehring et al. have recently described an elegant nuclear magnetic resonance (NMR) experiment implementing an algorithm to factor numbers based on the properties of Gauss sums. Similar experiments have also been described by Mahesh et al. In fact these algorithms do not factor numbers directly, but rather check whether a trial integer $\ell$ is a factor of a given integer $N$. Here I show that these NMR schemes cannot be used for factor checking without first implicitly determining whether or not $\ell$ is a factor of $N$.