Equations, codes, ciphers, mathematics and poetry
Michal Shurek says about himself: “I was born in 1946. I graduated from Warsaw University in 1968 and since then I have been working at the Faculty of Mathematics, Informatics and Mechanics. Scientific specialization: algebraic geometry. I recently dealt with vector bundles. What is a vector beam? So, the vectors need to be tightly tied with a thread, and we already have a bunch. My physicist friend Anthony Sim made me join the Young Technician (he admits he should be getting royalties from my fees). I wrote a few articles and then I stayed, and since 1978 you can read every month what I think about mathematics. I love mountains and, despite being overweight, I try to walk. I think teachers are the most important. I would keep politicians, whatever their options, in a heavily guarded area so they can't escape. Feed once a day. A beagle from Tulek likes me.
An equation is like a cipher for a mathematician. Solving equations, the quintessence of mathematics, is the reading of ciphertext. This has been noticed by theologians since the XNUMXth century. John Paul II, who knew mathematics, wrote and mentioned this several times in his sermons - unfortunately, the facts have been erased from my memory.
In school science, it is represented Pythagoras as the author of the theorem on some dependence in a right triangle. So it became part of our Eurocentric philosophy. And yet Pythagoras has much more virtues. It was he who imposed on his students the duty to "know the world", from "what is behind this hill?" before studying the stars. That is why Europeans "discovered" ancient civilizations, and not vice versa.
Some readers rememberViète patternsand"; many older readers remember the term itself from school and approximately the fact that the question appeared in quadratic equations. These regularities are “ideologically” encryption information.
No wonder one Francois Viet (1540-1603) was engaged in cryptography at the court of Henry IV (the first French king from the Bourbon dynasty, 1553-1610) and managed to break the cipher used by the British in the war with France. So he played the same role as the Polish mathematicians (led by Marian Rejewski), who discovered the secrets of the German Enigma cipher machine before World War II.
Fashion theme
Exactly. The topic "codes and ciphers" has long become fashionable in teaching. I have already written about this several times, and in two months there will be another series. This time I am writing under the impression of a film about the war of 1920, where the victory was largely due to the breaking of the code of the Bolshevik troops by a team led by the then young Vaclav Serpinsky (1882-1969). No, it's not Enigma yet, it's just an introduction. I remember a scene from the film where Józef Piłsudski (played by Daniil Olbrychski) says to the head of the cipher department:
The decoded messages carried an important message: Tukhachevsky's troops would not receive support. You can attack!
I knew Vaclav Sierpinski (if I may say so: I was a young student, he was a famous professor), attended his lectures and seminars. He gave the impression of a withered scholar, distracted, busy with his discipline and not seeing the other world. He lectured specifically, facing the blackboard, not looking at the audience ... but he felt like an outstanding specialist. One way or another, he had certain mathematical abilities - for example, for solving problems. There are others—scientists who are relatively bad at solving puzzles, but who have a deep understanding of the whole theory and are capable of initiating entire fields of creativity. We need both - although the first one will move faster.
Vaclav Sierpinski never talked about his achievements in 1920. Until 1939, this definitely had to be kept secret, and after 1945, those who fought with Soviet Russia did not enjoy the sympathy of the then authorities. My conviction that scientists are needed, like an army, is proven: "just in case." Here is President Roosevelt calling Einstein:
The outstanding Russian mathematician Igor Arnold openly and sadly said that the war had a great influence on the development of mathematics and physics (radar and GPS also had a military origin). I do not go into the moral aspect of the use of the atomic bomb: here is the extension of the war for a year and the death of several million of their own soldiers - there is the suffering of innocent civilians.
***
I run away to familiar areas - k. Many of us played with the codes, maybe scouting, maybe just like that. Simple ciphers, based on the principle of replacing letters with other letters or other numbers, are routinely broken if we catch only a few clues (for example, we guess the king's name). Statistical analysis also helps today. Worse, when everything is changeable. But the worst thing is when there is no regularity. Consider the code described in The Adventures of the Good Soldier Schweik. Take a book, for example, The Flood. Here are the suggestions on the first and second pages.
We want to encode the word "CAT". We open on page 1 and the next second. We find that on page 1, the letter K first appears in the 59th place. We find the fifty-ninth word on the opposite, the other side. It's an "a" word. Now the letter O. On the left is the 16th word, and the sixteenth on the right is "Mr." The letter T is in the 95th place, if I counted correctly, and the ninety-fifth word from the right is "o". So, CAT = 1 LORD O.
An "unguessable" cipher, albeit painfully slow both for encryption and ... for guessing. Suppose we want to pass the letter M. We can check if we encode it with the word "Wołodyjowski". And after us they are already preparing a prison cell. We can only count on a replacement! In addition, counterintelligence notes reports of secret employees that for some time customers have been willingly buying the first volume of The Flood.
My article is a contribution to this thesis: even the most bizarre ideas of mathematicians can find application in a broadly understood practice. For example, is it possible to imagine a less useful mathematical discovery than the test for divisibility by ... by 47?
When do we need it in life? And if so, it will be easier to try to separate it. If it divides, then it's good, if not, then ... secondarily it's good (we know that it doesn't divide).
How to share and why
After this introduction, let's move on to. Do you readers know any signs of divisibility? Definitely. Even numbers end in 2, 4, 6, 8, or zero. A number is divisible by three if the sum of its digits is divisible by three. Similarly, with the sign of divisibility by nine - the sum of the digits must be divisible by nine.
Who needs it? I'd be lying if I convinced the Reader that he was good for anything other than... school assignments. Well, and another feature of divisibility by 4 (and what is it, Reader? Maybe you will use it when you want to know what year the next Olympiad falls on ...). But the feature of divisibility by 47? This is already a headache. Will we ever know if something is divisible by 47? If yes, then take a calculator and see.
This. You are right, Reader. And yet, read on. Please.
Divisibility by 47: The number 100+ is divisible by 47 if and only if 47 is divisible by +8.
The mathematician will smile with satisfaction: "Gee, pretty." But mathematics is mathematics. Evidence matters, and we pay attention to its beauty. How to prove our trait? It's very simple. Subtract from 100 + the number 94 - 47 = 47 (2 -). We get 100+-94+47=6+48=6(+8).
We have subtracted a number that is divisible by 47, so if 6 (+ 8) is divisible by 47, then so is 100 +. But the number 6 is coprime to 47, which means that 6 (+ 8) is divisible by 47 if and only if it is + 8. End of proof.
let's get a look Some examples.
8805685 is divisible by 47? If we're really interested in it, we'll find out sooner just by dividing us like we were taught in elementary school. One way or another, now there is a calculator in every mobile phone. Divided? Yes, private 187355.
Well, let's see what the sign of divisibility tells us. We disconnect the last two digits, multiply them by 8, add the result to the “truncated number” and do the same with the resulting number.
8805685 → 88056 + 8 · 85 = 88736 → 887 + 8 · 36 = 1175 → 11 + 8 · 75 = 611 → 6 + 8 · 11 = 94.
We see that 94 is divisible by 47 (the quotient is 2), which means that the original number is also divisible. Fine. But what if we keep having fun?
94 → 0 + 8 94 = 752 → 7 + 8 52 = 423 → 4 + 8 23 = 188 → 1 + 8 88 = 705 → 7 + 8 5 = 47.
Now we must stop. Forty-seven is divisible by 47, right?
Do we really need to stop? What if we go further? Oh my God, anything can happen ... I will omit the details. Maybe just the beginning:
47 → 0 + 8 · 47 = 376 → 3 + 8 · 76 = 611 → 6 + 8 · 11 = 94 → 0 + 8 · 94 = 752.
But, unfortunately, it is as addictive as chewing seeds ...
752 → 7 + 8 * 52 = 423 → 4 + 8 * 23 = 188 → 1 + 8 * 88 = 705 → 7 + 8 * 5 = 47.
Ah, forty-seven. It happened before. What's next? . Same. The numbers go in a loop like this:
It's really interesting. So many loops.
Two following examples.
We want to know if 10017627 is divisible by 47. Why do we need this knowledge? We remember the principle: woe to knowledge that does not help the knower. Knowledge is always there for something. It will be for something, but now I will not explain. A few more accounts:
10017627 → 100176 + 8 27 = 100392.
"He changed his uncle from an ax to a stick." What do we get from all this?
Well, let's repeat the course of the proceedings. That is, we will continue to do this (that is, the word “iterate”).
100392 → 1003 + 8 92 = 1739 → 17 + 8 39 = 329 → 3 + 8 29 = 235.
Let's stop the game, divide like in school (or on a calculator): 235 = 5 47. Bingo. The original number 10017627 is divisible by 47.
Well done!
What if we go further? Trust me, you can check it out.
And one more interesting fact. We want to check if 799 is divisible by 47. We use the divisibility function. We disconnect the last two digits, multiply the resulting number by 8 and add to what is left:
799 → 7 + 8 = 99 + 7 = 792.
What we have? Is 799 divisible by 47 if and only if 799 is divisible by 47? Yes, that's right, but no math is needed for this!!! The oil is oily (at least this oil is oily).
About the leaf, pirates and the end of jokes!
Two more stories. Where is the best place to hide a leaf? The answer is obvious: in the forest! But how can you find it then?
The second we know from books about pirates that we read a long time ago. The pirates made a map of the place where they buried the treasure. Others either stole it or won the fight. But the map did not indicate which island it was intended for. And look for yourself! Of course, the pirates coped with this (torture) - the ciphers I'm talking about can also be extracted using such methods.
End of jokes. Reader! We create a cipher. I'm an undercover spy and use "Junior Technician" as my contact box. Forward me encrypted messages as follows.
First, convert the text to a string of numbers using the code: AB CDEFGH IJ KLMN ON RST UWX Y Z1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
As you can see, we don't use Polish diacritics (i.e. without ą, ę, ć, ń, ó, ś) and non-Polish q, v - but the non-Polish x is there just in case. Let's include another 25 as a space (space between words). Oh, the most important thing. Please apply code no. 47.
You know what that means. You go to a friend mathematician.
The friend's eyes widened in surprise.
You answer proudly:
A mathematician endows you with this trait... and you already know that an inconspicuous-looking function is used for encryption
because such a pattern is a described action
100+→+8.
So, when you want to know what a number means, like 77777777 in an encrypted message, you use the function
100+→+8
until you get a number between 1 and 25. Now look at the explicit alphanumeric code. Let's see: 77777777 →… I leave this to you as a task. But let's see what letter 48 hides? Let's read:
48 → 0 + 8 48 = 384.
Then we get in turn:
384 → 3 + 8 84 = 675 → 6 + 8 75 = 606 → 6 + 8 6 = 54 → 0 + 8 54 = 432 ...
The end is not in sight. Only after the sixtieth (!) time will a number less than 25 appear. This is 3, which means 48 is the letter C.
And what does this message give us? (I want to remind you that we use code number 47):
80 - 152 - 136 - 546 - 695719 - 100 - 224 - 555 - 412 - 111 - 640 - 102 - 152 - 12881 - 444 - 77777777 - 59 - 408 - 373 - 1234567 - 341.
Well, think about it, what's so complicated, some accounts. We have begun. Early 80. Known rule:
80 → 0 + 8 80 = 640 → 6 + 8 40 = 326.
It continues like this:
326 → 211 → 90 → 720 → 167 → 537 → 301 → 11.
Eat! The first letter of the message is K. Phew, easy, but how long will it take?
Let's also see how much trouble we have to have with the number 1234567. Only on the sixteenth time will we get a number less than 25, namely 12. So 1234567 is L.
Okay, one might say, but this arithmetic operation is so simple that programming it on a computer will break the code right away. Yes it's true. These are simple computer calculations. idea with public cipher and it is also about making the calculations difficult for the computer. Let it work for at least a hundred years. Will he decrypt the message? Does not matter. It won't matter for a long time. This is (more or less) what public ciphers are about. They can be broken if you work for a very long time ... until the news is no longer relevant.
it has always given birth to "counterweapons". It all started with a sword and shield. The Secret Services pay huge sums of money to gifted mathematicians to invent encryption methods that computers (including those created by us) will not be able to crack in the XNUMXth century.
twenty-second century? It is not so difficult to know that there are already many people in the world who will live in this beautiful century!
Oh huh? What if I ask (me, the Secret Officer contacted by the “Young Technician”) to encrypt with code number 23? Or 17? Simple:
May we never have to use mathematics for such purposes.
***
The title of the article is about poetry. What does she have to do with this?
Like what? Poetry also encrypts the world.
How?
By their methods - similar to algebraic ones.
