Factoring[8 616 460 799] like 100 years ago

[C.Ret]

(09-06-2022 04:51 PM)Thomas Puettmann Wrote:  You can exclude a number from beeing a square by inspecting its last digit. If the last digit is 2, 3, 7, or 8 then the number is not a square.
The factoring machine performs precisely this test. Not in the decimal system but simultaneously in the 25, 36, ..., 31 number systems.
I assume in the following that the machine should factorize the integer 8 616 460 799.
To configure the chain with 36 links for example, you start at 0 in the counter and crank up to 36. At each number \( n \) in this range you put a "track link" onto the top chain link if the last digit of \( 8 616 460 799 + n^2 \) is not that of a square in the 36 number system (i.e. mod 36).
The machine stops if the last digits of \( 8 616 460 799 + n^2 \) is that of a square in all the number systems above. Then the laser beam can pass without obstruction to the photo transistor. You then have to check manually whether \( 8 616 460 799 + n^2 \) actually is a square integer. If it is not, you let the machine continue. If yes, you factor the number as shown in the video.
[...]
You can't make the length of a chain shorter than 20 links. So you use 22 instead of 11, for example.

Thank you very much for these very clear and detailed explanations.

I now understand better why these numbers are not simply prime factors; I hadn't thought of any mechanical constraint. Indeed the simplest common and practical sense; the chains must be longer than the wheel that drives them...
...sometimes the simplest and most pragmatic constraints are the hardest details to pin out.

This is also the point of making such a machine yourself. The realization is certainly easier by using construction sets (Fishertechnics, Lego, Meccano, ...) than building everything from scratch and having to manufacture and design every piece by yourself.

Otherwise, I also wanted to congratulate you on the video which is very well made; good video quality (especially the light is excellent - did you use a white glove box?)
As well as the idea of ​​not commenting in one language or another which makes it accessible to everyone. Now knowing the reason for these eight numbers at the bottom of each channel, I understand better why it could not be effectively shown in the video in a reasonably short time.

I also like the added video after the end which explains using a hand's watch why it should read 3199 and not 3208. It is by building such a system that we understand the limits and discovers the imperfections or shortcomings of any technology and mechanics. This is a very good educational idea.

I hope that the number of viewings will considerably increase and especially that your book will sell decently and will awaken many young vocations.

Anyway, this all piqued my curiosity and I now have a few ideas to try to speed up this factorization on my poor, old and slow Ti-58c.