Factoring[8 616 460 799] like 100 years ago

[Thomas Puettmann]

Quote:Indeed, what is the trick and how the number 8'616'460'799 is encoded in the strings?

A brief answer to this question is given in my reply to Thomas Klemm's comment. I will try to explain it in more detail:

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.

Quote:Why this curious series of numbers { 25 36 21 23 22 29 23 31 }
Why aren't they all primes?

You can't make the length of a chain shorter than 20 links. So you use 22 instead of 11, for example.

Additionally, you also want to use powers of primes like in 36 = 4 · 9. For example, the last digit of every binary number is that of a square number (0 or 1). On the other hand, numbers that end on 3 in the 4 number system can be excluded from beeing squares.

In addition to the large factoring machine in the video I also built a minimal machine with only two chains (with 28 and 45 links) and much fewer parts. It is ideally suited to learn this kind of factorization. A video with explanations will follow at some point of time on my channel.

Quote:There are many mysteries that will have to be explained to us.

I am glad that the video brought these questions to you. From my perspective, raising questions is more important than providing answers.

Quote:... Unless all this is just a simple publicity campaign for an upcoming book?

The main purpose of the video is the presentation of the machine. The hint to the book is just a side purpose. If I composed this video for "publicity campaigning for an upcoming book", it would be a total failure, since according to the YouTube statistics almost all of the so far 230 viewers already quitted viewing after 1:45 "task completed".

The vast majority of forum members cannot read German and never had any experience with fischertechnik. As EdS2 puts it in his post "... I could have been a fan of Fischer-Technik if I’d had enough of it…" So, it doesn't make much sense to show the video here in order to sell the book.

What I do appreciate is the feedback for the machine and for my somewhat "nerdy" project/mission "learn math with mechanical models" in general. I deeply believe that every child should be given the opportunity to build a mechanical calculator by itself in order to really understand the concept of a "carry".