SYRACUSE : LA PREUVE À 850 000 DOLLARS !

Aujourd’hui, nous allons parler d’un mystère dans le monde des maths. Une conjecture qu’un enfant de 8 ans pourrait comprendre, et qui pourtant, reste non prouvée à ce jour. Embarquez avec nous, attachez vos ceintures, et préparez vous pour un voyage passionnant vers la conjecture de Syracuse!

► POUR SOUTENIR LA CHAÎNE :
• S’abonner : https://www.youtube.com/channel/UCxX0XtLEus9Hu-PawWgtbXQ?sub_confirmation=1
• Un petit pourboire?
https://fr.tipeee.com/mathador
• Partager cette vidéo à vos amis
• Mettre des pouces bleus, et poster des commentaires
► RÉSEAUX SOCIAUX
• facebook : https://www.facebook.com/Mathador-960047670809006
• twitter : https://twitter.com/mathador5
• Instagram : https://www.instagram.com/franckdunas

► Les chaines des copains qui interviennent dans cette vidéo :
• Le museum des pourquois : https://www.youtube.com/@LeMuseumdesPourquois
• e-penser : https://www.youtube.com/@Epenser1
• Science de comptoir : https://www.youtube.com/@Sciencedecomptoir
• le point genius : https://www.youtube.com/@LePointGenius (pas en voix off, mais PH m’a aidé pour un plan d’incrustation sur un avion à la fin) Merci mon PH
► SOURCES :

Enoncé rapide : https://www.maths-et-tiques.fr/index.php/detentes/la-conjecture-de-syracuse

Le site de Gerard Villemin, une mine d’or, et source régulière pour mes vidéos : http://villemin.gerard.free.fr/Wwwgvmm/Iteration/Syracuse.htm

l’excellente chaîne @ElJj qui a consacré un épisode entier sur cette conjecture :

La page de blog de David Louapre. Ai-je besoin de le présenter?

La conjecture de Syracuse

Loathar Collatz : https://fr.wikipedia.org/wiki/Lothar_Collatz

La publication de Richard Crandall : https://www.ams.org/journals/mcom/1978-32-144/S0025-5718-1978-0480321-3/S0025-5718-1978-0480321-3.pdf

L’avancée de Terence Tao : https://www.larecherche.fr/mathématiques/un-peu-plus-près-de-syracuse
https://arxiv.org/pdf/1909.03562.pdf

Un site très interessant, qui résume notamment l’analogie avec la musique : http://www.probleme-syracuse.fr/math.html

Un simulateur. Vous entrez un nombre de départ, ça vous donne tous les termes de la suite de Syracuse! https://calculis.net/syracuse

Quelques articles de recherche sur la conjecture :
• https://sci-hub.3800808.com/10.1090/S0002-9939-1981-0603593-2
• https://sci-hub.3800808.com/https://link.springer.com/chapter/10.1007/978-3-540-72504-6_49
• https://sci-hub.3800808.com/10.1007/s002360050117
• https://sci-hub.3800808.com/10.1007/s11227-020-03368-x

► La chaîne Mathador est membre du café des sciences : https://www.cafe-sciences.org/

Hello! Hello. Ah, I see. This is your place, isn’t it? Yes, it is. 28 B. Sorry about that, I’ll have to disturb you. That’s fine. Please go ahead. Sorry, sorry, sorry… There you are! I really do nowadays,

The more things go, the less leg room legroom on planes! Yes, and still, I have the possibility to stretch them out a bit in the aisle

That’s true, but I’m not going to complain. But I’m not going to complain I’ve got a window and a view of the clouds! Wow! Don’t shout victory too quickly We haven’t taken off yet!

Ha ha ha! Well, we don’t know each other yet, but I’ve got the feeling that you’re are a great optimist, you! It’s a joke, of course. Nice to meet you. My name is Walter. Nice to meet you. My name is Franck.

I looked at the weather forecast: we’re lucky It’s bright and sunny in New York… Just as well! Because I have to admit I’ve remained very modest about the amount of jumpers in my suitcase. And tell me: what are you going to do in New York?

An acquisition. Because you see, I’m in charge a museum. The “Museum of Why”, and that… Good morning ladies and gentlemen, this is the captain the captain. You are now seated on flight 142. Air Syracuse, bound for New York. Our flight time today will be

Of 8h04min. We can count on good weather during our journey despite some light turbulence. We land at 13:40 local time at Kennedy Airport. The whole team wishes you a pleasant journey with Syracuse.

Syracuse… Strange, isn’t it, this airline? Do you think it has something to do with the city? I think it’s to do with the conjecture. The? The Syracuse conjecture. Don’t you know it? No, I haven’t. It doesn’t ring a bell! It’s quite a crazy story. Would you

To hear it? I’d love to! It all began in 1937, with this man : Lothar Collatz. This is a German mathematician, working at… Wait, wait… Reassure me: you’re not you’re not going to talk to me about mathematics,

Here? No, because… how can I put this… We have a flight of over 8 hours, and I might as well tell you straight away that I’m not I’m not at all keen on maths!

Don’t worry: in everything I’m about to tell you, there’s no I’m about to tell you, there’s nothing that a child of wouldn’t understand. Oh, right. Sorry about that. Carry on, then! Lothar Collatz works as an assistant at the University University of Berlin as an assistant

D. in 1935. A remarkable work on linear differential equations. But that’s not what we’re interested in here. Because to relax, Collatz is in the habit of think about a small problem, which circulates in the world of mathematicians for some time now.

Years. A sort of enigma, which is passed around under the table. And here it is: Take a number, any number. If it is even, divide it by 2. If it’s odd, take its triple, and add 1. You will then find another number. Repeat this operation,

Again and again. You will invariably invariably on the number 1, with this infinite loop: 1,4,2,1,4,2, etc… Question: Even if this that we always come back to this loop seems to be true for the first numbers,

Can we state definitively that the same is true for whatever the starting number? Collatz thinks so. He then stated this conjecture in 1937: Whatever be the strictly positive number, the sequence consisting of the terms from this simple

Rule will invariably lead to the cycle 1,4,2,4,1,4,2 etc…, In 1952, Collatz shared his conjecture with Helmut Hasse, one of the world’s greatest mathematicians. German algebraists of his time, who the university in a small town.

City in the state of New York: Syracuse Syracuse University. And then, very quickly, this strange and seemingly simplistic problem spreads like wildfire. all over the United States, a conjecture which the name of this university

Of which this mathematical earthquake was epicentre: The Syracuse conjecture. If it quickly fascinated mathematicians of the 2nd half of the 20th century. 20th century, it was not only because its simplistic simplistic statement contrasts with the difficulty

They encountered in proving it, but also because studying the terms of the Syracuse sequence reveals some very interesting interesting specificities and questions. to which researchers have been quick to quickly wanted to find answers.

For example, depending on the starting number chosen, the graphical representations show that, unsurprisingly, all the sequences end up reach the number 1, but in a time time. The Syracuse sequence for N=6 (i.e., the sequence of numbers obtained

Starting with the number 6) requires 9 results before 1: 6, 3, 10, 5, 16, 8, 4, 2, 1 Whereas for N=7, you have to wait for 17 results: 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

This time is called the “time of flight” by mathematicians. mathematicians. To be more precise, if we the terms of the sequence, then the time is the smallest index n such that Un=1. From there, a whole aviation vocabulary was born to complete this description. For example,

If the conjecture is true, then in a sequence, whatever the starting number, there will necessarily be a maximum number. It is 160 for the Syracuse sequence of 15, or 16 in the Syracuse sequence of 5. This number is called the maximum altitude.

Finally, we can also look at the time it will take it will take for the sequence to descend below of its starting term. For example, if you start from 11, the next 7 terms will be greater than

11, and it is only at the next term, 10, that the sequence will become less than our starting number. (11 , 34 , 17 , 52 , 26 , 13 ,40 , 20 , 10 , 5 , 16 , 8 , 4 , 2 ,1). This

Time is called the altitude flight time. Sometimes this flight time at altitude is very long: For example, it is 96 for the Syracuse 27 suite. Or very short : only 2 in the Syracuse 34 sequence (34 ,

17 , 52 , 26 , 13 ,40 , 20 , 10 , 5 , 16 , 8 , 4 , 2 ,1). Fascinating. So we can say that each departure number has a different flight, right?

That’s right. And each number of starts has its characteristics, and records, too! Records which are not linked to the height of the starting start number, by the way. Interesting stuff. For example, between 0 and 100, you might think 99 has the highest maximum altitude.

When in fact it’s the number 27. Or, between 0 and 1000, it’s the number 871 which has the longest flight time. But then, this conjecture… has anyone proved it, or not? Ahhhh… the proof of Syracuse…

Good morning ladies and gentlemen, this is the captain the captain. As planned, we’re going to pass through a turbulence. Please fasten please fasten your seatbelts during this period of turbulence. The Syracuse conjecture is based on the quantifier “Whatever”. Therefore,

There is a fairly simple principle in mathematical logic to disprove an assertion based on on a “whatever”: It is necessary and it is just find a counterexample. To illustrate, let’s take the assertion: “Whoever is on that plane,

He is right-handed”. To prove that this sentence is false, you don’t need to ask each passenger if they’re right-handed: all you have to do is find a left-handed person among us to prove that was wrong.

It’s the same thing with the conjecture of Syracuse conjecture, which says: “Whatever the number n of the Syracuse sequence (Un) necessarily falls to to 1, and all the more so in the infinite cycle: 1,4,2…

All you have to do is find a single starting number that doesn’t fit, and it’s a done deal. But here’s the first problem on closer inspection closely: Even if the time of flight of a number is large enough for us to

Suspect that we won’t get back to 1… the list of successive operations is infinite, and there’s nothing to say that if we continue far enough far enough, we won’t end up back on 1… For

Be able to state categorically that a number generates a sequence that will NEVER end up at 1, it would therefore be necessary to be able to examine an infinite number of iterations. This is impossible.

Impossible, unless we can show that the sequence generated by the starting number diverges, i.e. it explodes towards infinity, without ever coming back down. In other words, find a starting number that generates an infinite flight time.

Another method of showing that the conjecture would be false: find a starting number that generates an infinite cycle cycle other than 1,4,2. For example, if a start number led to the cycle a,b,c, a, b, c etc., then by definition

The sequence would never return to the number 1, and therefore the conjecture would be false. But this is far from being the position of the researchers. Because at the moment I’m talking to you in 2023,

The conjecture has been verified by computer to a starting number of 2.95.10^20, i.e. 295 followed by 18 zeros! So we have every reason to to think that the conjecture is true. But in Mathematics, a proof is a proof,

And the word “conjecture” will only be replaced by theorem only when a solid, verified and validated by the community. I imagine that since 1952 progress in this direction, don’t you think?

There’s no proof. But progress, of course. For for example, there is a probabilistic argument which really suggests that the sequence will necessarily decrease over time. Would you like me to explain? Is it easy to understand? Very simple, you’ll see:

First of all, what is an even number? It’s simple, as we learned very early on at at school: it’s a number that can be divided by 2, i.e. a number that equals

Twice another number. For example: 8 is even because you can write it as 2*4. Another example is 26 is even because you can write it as 2*13. Generally speaking, this is how you write

Write an even number: A number that can be written as “2* ‘another number'”. So what is an odd number? Well, in the simplest possible way, an odd number is the following an even number. So in Maths, to write an odd number,

We take an even number (2*one number) and add 1 (2*one number +1). Now that we know this, let’s get back to our to our Syracuse sequence. There are some particularly interesting to notice in its statement: If,

Starting from an odd number, we perform its triple by adding 1, you are bound to end up with an odd number. Here’s why: Let’s start with any odd number. If take a closer look at the wording:

3 * the number + 1, we can then write by saying that it’s 2 * the number + 1 + the number. What do we notice then? That we are doing the sum of two odd numbers the sum of 2* the number + 1,

Which is precisely the definition we have seen of an odd number, with the starting number, which, was odd. So what happens when you add two odd numbers? You invariably end up with an even number. All this to say that in the Syracuse sequence,

If the number is odd, then its successor is necessarily even, and that in the operation we’ll have to divide it by two. Some mathematicians have therefore compiled this result to update the Syracuse sequence in this way:

If the starting number is even: divide by two divide by two (nothing changes here) On the other hand, if it is odd: Then take the successor of the triple, as before, and divide it directly by 2. This formulation

Of the Syracuse sequence is called the “compressed” form. Wait, wait, but… what’s the point? We would have halved it anyway in the next operation, wouldn’t we? I don’t quite the point of this “compressed form”.

I understand your reservations. Let me show you show you why it’s clever: To sum up: If the start number is even: we divide it by two, i.e. multiply it by 1/2. If it is odd, the calculation gives (3 * the number +1)/2. Ignoring the 1,

We can see that in this case, we multiply the starting number by approximately 3/2. By repeating the sequence from Syracuse sequence to infinity, we multiply by 1/2, or multiply by 3/2, with a probability of a probability of 1/2, since we have as many

Chance of finding an even number as an odd one. In other words, on average, we will successively multiply by (1/2)^0.5*(3/2)^0.5 = 0.866… Hmm… I think I see what you’re getting at… 0.866 is smaller than 1. Now

Everyone knows that when you multiply by a number smaller than 1, that number decreases… And so, by successive by 0.866, the starting number becomes in a way condemned to constantly decrease. Absolutely. And it’s this decrease that

Suggest that the arrival on the number 1, and therefore the 1,4,2 cycle, is inexorable. Unfortunately, this argument is only a probabilistic probabilistic clue, and is not valid as a proof. Prestigious mathematicians have produced brilliant brilliant work, such as Richard

Crandall, who estimated that the ratio between two odd terms is about 3/4, or Terence Tao, one of the greatest mathematicians of our time, who made a historic breakthrough in 2019. But then again,

The arguments developed will not allow a proof of the conjecture, which seems to resist all the assaults the greatest brains of this century. But why is this seemingly simple so simple so complex?

Go figure. Maths magic, I imagine. And maybe also the fact that that this problem occurs in other like music, for example. music? Why is that? Because in music, the ratio separating a note from its lower octave octave is 1/2. And the ratio between

A note from its fifth is 3/2. We find Syracuse values. But that’s not all, because researchers soon realised that this suite suite offered much wider areas of study. varied than the simple little game we started with. Did they? And why is that?

“Why?” I get the impression that you particularly like that word, don’t you? Yes, indeed, it’s an adverb that I’m familiar with… In the end, the Syracuse conjecture seems to be the Syracuse conjecture seems to

The most brilliant mathematicians since it was formulated by Lothar Collatz, who, dying in 1990, never saw the solve his problem. At its big sisters, the conjectures of or the Riemann hypothesis, it has stood the test of time and the assaults of the greatest minds.

With the difference that its very simple wording attracted many more fans, who, in an excess of confidence think that if the statement is simplistic, then perhaps the solution is within their grasp. That’s how hundreds of people get

Have ventured to propose demonstrations with various motivations, ranging from the good faith to more dubious motives: for some, highlighting the for some, the lure of a good price. for others (the Bakuage company offering a price of 120 million yen for its resolution).

Sometimes the attempts are commendable, but it’s clear that when you from a mathematician’s profile, the content of the the content of the proposed demonstrations more of a science… how how shall I put it… science…

… De comptoir. That’s the word you’re looking for, sir. By the way, would you like a drink? Yes, please. I’ll have a doctor doctor doctor peper No problem. Well struck, I guess? The fact remains that the mathematical community is not giving up. It’s

Far from being the first conjecture that mathematicians, but time is running out. works for them. We saw it with Fermat’s conjecture proved in 1995 by Andrew Wiles, Poincaré’s conjecture, demonstrated in 2003 by Perelman The demonstration often ends up happening,

And there is no doubt that as soon as a real proof is provided of Syracuse, the list of mathematical theorems will increase by one line. But in the meantime, the problem remains open. This is the captain speaking, our flight is now arriving at its destination,

After a flight time of 30 and a maximum altitude of 592 (NB: On screen: a mathador T-shirt offered to the first person to find the start number corresponding to these characteristics). We are hoping for

That you have had a pleasant journey journey with Syracuse. Well, it remains for me to say goodbye and wish you a pleasant stay. Thank you for your time. And you too! Perhaps we’ll have the opportunity to meet again one of these days?

You’re not quite right. Because in my museum, I intend to talk about all unresolved problems, the so-called “open” problems, where the Syracuse conjecture is obviously part of it. I would be delighted to to talk about it together!

Listen, it would be a great pleasure. Wonderful. I can’t wait! I can’t wait either. So see you at the museum des pourquois!