# Continued Fractions Applied to the One Line Factoring Algorithm for Breaking RSA
## 8 Conclusions  
This paper proposed a new approach of applying continued fractions to the one line factoring algorithm. Lower ordered convergences as representations of \(\frac{p}{Q}\) that form continued fractions were applied to [PERSON]'s one line factoring algorithm. We illustrated our method using a worked example as well as by finding the factors of RSA250 successfully. The key contributions of this work that paves the way for further research are as follows:  
1. We established that continued fraction convergences \(\frac{p}{q}\) as representations of \(\frac{p}{Q}\) provide solutions to [PERSON]'s one line factorization.
2. We explored relationships of the form \(kN=x^{2}-y^{2}+z^{2}\) for convergence.
3. We investigated the suitability of the modified Brahmagupta-Fibonacci identity for factorization considering \[kN=a^{2}-b^{2}-c^{2}+d^{2},\;ad-bc=0\qquad kN=(a-d)^{2}-(b-c)^{2}\]
4. Our experimental findings include the RSA modulus, the ratio of \(\frac{p}{Q}\) to be constrained by a particular number of bits. \(P,Qbits=\frac{N}{2}bits\Rightarrow 1<\frac{p}{Q}<2\;1<\frac{p}{q}<2\) such that \(x^{2}-y^{2}=4 kN,k=pq\). \(k\) is still an area of interest and may yet yield deeper connections.  
In this paper, we studied many correct solutions which formed the convergences of \(PQ\) that had not been explored previously. The results shown in this paper are quite important as we explored the relationship between the convergence of \(PQ\) and its relationship to \(k\), which is used in [PERSON]'s one line factorization algorithm. Our significant contribution is that after about one third of the way through the convergences, solutions for \(k\) were found and all convergences after that resulted in valid solutions for \(k\). Further, there is a scarcity of research exploring the attributes of \(k\), and this paper provides a better understanding of its attributes.  
Further, our practical implementation along with our findings motivated us to extend the study using the difference of squares. The theory on the difference of squares provides the relationship to the proposed approach and future extensions to our algorithm, motivating other researchers to consider such extensions to the existing work. We believe the observations and limitations of this will open future investigations as forms of ongoing research.  
In Section 4, we showed that the convergence that provides a value for \(k\) can be bound to a search field of composites with reduced complexity to about \(1/3\) of the number attempting to be factorized whilst acknowledging that this is now a search in a composite field rather than a search in a prime field. In Section 6, the attributes of \(k\) were further defined, thereby constraining the suitable composite candidates for \(k\).  
Our findings are unique in that only one other paper has been published regarding [PERSON]'s one line factoring algorithm since its publication in 2012.  
It is hoped that the wider research community might find our results interesting and that further investigations into [PERSON]'s work might be encouraged.  
Conceptualization, A.O.; methodology, A.O. and S.V.; formal analysis, A.O.; validation, S.V.; investigation, A.O. and S.V.; resources, A.O.; data curation, A.O.; writing--original draft preparation, A.O.; writing--review and editing, S.V. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The data presented in this study are available on request from the corresponding author. The authors declare no conflicts of interest.