Confusion on the original Lucas test

lucas-numbersnumber theoryprimality-test

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 using publications from the authors and various sources – some of which I'll leave below, more on them in the appropriate paragraph.

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 is 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 Lucas did not properly formulate it, he just used it in some confused example and left it there. Still, I could not find a great deal of info. 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, even 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, it's my first on the site!

Duplicate(?)

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 siuation.

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.

Useful sources

The original article can be found in French on JSTOR, divided into two parts:

  1. American Journal of Mathematics , Vol. 1, No. 2 (1878), pp. 184-196. http://www.jstor.org/stable/2369308
  2. 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.

  1. 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.
  2. [?]

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.

  1. R. D. Carmichael. “On the Numerical Factors of the Arithmetic Forms $αn±βn$”, Annals of Mathematics15 1/4 (1913), pp.30–48. http://www.jstor.org/stable/1967797.
  2. R. D. Carmichael. “On the Numerical Factors of the Arithmetic Forms $αn±βn$”, Annals of Mathematics15 1/4 (1913), pp. 49–70. http://www.jstor.org/stable/1967798.

Best Answer

User @Conifold from History of Science & Mathematics stackexchange provided a perfect source, I shall briefly report the information it contains to close the question. I hope answering myself is fine in these situations, feel free to correct or post another answer. This is a repost from the discussion in HSM.se.

See Dickson's History of the theory of numbers, chapter XVII for the full content.

After defining his sequences $u_n$ and $v_n$, Lucas studied relations between their terms and gave some properties of purely algebraic nature. These were then properly demonstrated.

What I was searching for can be found starting from page 397:

Lucas stated that, if $4m+3$ is prime, $p = 2^{4m+3} — 1$ is prime if the first term of the series 3, 7, 47,..., defined by $r_{n+1} = r_n^2 — 2$, which is divisible by $p$ is of rank $4m+2$; but $p$ is composite if no one of the first $4m +2$ terms is divisible by $p$.

This will of course be later reformulated in the famous Lucas-Lehmer test. The formulation Lucas gave, albeit weaker than Lehmer's, can find some divisors if the number is composite.

Lucas then proceeds giving some examples: see page 399 for a primality test for numbers in the form $p = 2^{4q+1} — 1$, note that more can be found. Important paragraphs are conveniently marked with a pencil.

The chapter then proceeds with more applications and examples of historical relevance, like exchanges with other mathematicians of the time.

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