I am currently researching on all primality tests deriving from Lucas' original paper Théorie des Fonctions Numériques Simplement Périodiques, which is of course known for its great deal of confusion. Still, I managed to make my way trough basically all of the recent tests (LLehmer, LLehmerRiesel, etc.) using publications from the authors and various sources - some of which I'll leave below, more on them in the appropriate paragraph. Note that there are various revisions from Lucas himself of the article, but every work of Lucas is a perfect source to me. I will not differentiate between them.
Back to us: time came I had to write about the original Lucas test in order to further develop it into the Lucas probable prime test and then the Bailie-PSW test. Of course those two are of no concern; there are plenty of sources. However I hardly managed to find something about the original formulation Lucas gave; lots of tests have the name "Lucas" in them, but I'm interested in that original one. It is my understanding that Lucas did not properly formulate it, but he just used it in some confused example and left it there. Still, I could not find a great deal of info.
So in short I'm trying to understand what Lucas did on its original paper - no reworks as in Carmichael; I am interested in the original for historical reasons. The article is known to be kind of a mess, and what's more it's in French -- I can't understand it nor find a translation. I do not know what version of the test he used; from what I read I suppose it was just a one-time procedure on an example which was then further developed in most of the "Lucas" tests. Specifically, my question is
- does anyone have a better idea on how he exactly used his sequences in primality testing on the original paper?
- are there some examples, especially from his paper, you could share?
- are there some better sources? of every kind: papers, videos, etc...
Thank you all in advance and sorry for the long post / weird formatting, it's my first time on SE!
While this is a well discussed topic, I hardly found discussions about the original test by Lucas and I thought it would be useful for the community to put here all my sources and ask for more insight on the original work - not the latter ones. That I did not find, and I'd like to help out people in my situation.
For example, I encountered this discussion which gave a book by Hans Riesel itself as an interesting source. Fact is, I do not have access to that book.
I also posted this same question on Mathematics stackexchange, and was advised to cross-post it here due to its historical focus. I modified my post a little since it seemed kinda ambiguous on the nature of the test I'm interested in.
The original article can be found in French on JSTOR, divided into two parts:
- American Journal of Mathematics , Vol. 1, No. 2 (1878), pp. 184-196. http://www.jstor.org/stable/2369308
- American Journal of Mathematics , Vol. 1, No. 4 (1878), pp. 289-321. http://www.jstor.org/stable/2369373
and I leave for posterity an English translation and revision of the first part -- it could be of great help for someone. If anyone manages to find the second part I'll be glad to add it too -- I couldn't and I'm not sure it does exist.
- Sidney Kravitz, Douglas Lind: “The theory of simply periodic numerical functions”, Fibonacci Association (1969). https://www.mathstat.dal.ca/FQ/Books/Complete/simply-periodic.pdf.
Last but not least, the famous Carmichael did a great job revising the original article; sadly he modified it to make it work better, which is awesome but not what I need. As before, I leave it here for posterity - it's a life saver.
- R. D. Carmichael. “On the Numerical Factors of the Arithmetic Forms $α^n±β^n$”, Annals of Mathematics15 1/4 (1913), pp.30–48. http://www.jstor.org/stable/1967797.
- R. D. Carmichael. “On the Numerical Factors of the Arithmetic Forms $α^n±β^n$”, Annals of Mathematics15 1/4 (1913), pp. 49–70. http://www.jstor.org/stable/1967798.