What is a conjecture in mathematics?

1 – Very first numbers … or not!

Consider, for each natural integer n, the integer: conjecture

    \[ Q_{n}=n^{2}+n+41\]

We can easily calculate: conjecture

    \[ \begin{matrix}Q_{0} & = & 0^{2}+0+41 & = & 41\\ Q_{1} & = & 1^{2}+1+41 & = & 43\\ Q_{2} & = & 2^{2}+2+41 & = & 47\\ Q_{3} & = & 3^{2}+3+41 & = & 53\\ Q_{4} & = & 4^{2}+4+41 & = & 61\end{matrix}\]

And so on \ldots{}

With a little habit, something strange challenges us …

These numbers are all first.

Recall that a natural integer is said to be prime if it has, in all and for all, two divisors: 1 and itself. Thus, for example, 13 is prime: its only divisors are 1 and 13. But this is not the case of 21, which is certainly divisible by 1 and 21, but also by 3 and 7. Prime numbers less than 30 are:

2,3,5,7,11,13,17,19,23,29

and the list goes on … indefinitely! About 23 centuries ago, the great Greek mathematician Euclid inscribed in his Elements the first demonstration of the infinity of prime numbers. conjecture

Fundamental fact: any integer greater than 1 is uniquely expressed (to the order of factors) as the product of prime numbers. For example : conjecture

    \[ 123\thinspace456\thinspace789=3\times3\times3607\times3803\]

Prime numbers can not be decomposed as the product of smaller numbers, and serve as elementary bricks to decompose all other integers. The prime numbers are in a way the “atoms” of the set of positive integers, provided with its multiplicative structure. conjecture

To learn more about them, you can read “Prime Numbers” from Wikipedia (to start).

Let’s go back to the numbers Q_{n}\text{\ldots} We saw that Q_{0},Q_{1},Q_{2},Q_{3} and Q_{4} are first. After that ?

To be clear, let’s continue the exploration a little further …

    \[ \begin{matrix}Q_{5} & = & 5^{2}+5+41 & = & 71\\ Q_{6} & = & 6^{2}+6+41 & = & 83\\ Q_{7} & = & 7^{2}+7+41 & = & 97\\ Q_{8} & = & 8^{2}+8+41 & = & 113\\ Q_{9} & = & 9^{2}+9+41 & = & 131\end{matrix}\]

You can check (if the heart tells you): these five are all first, too! And you can even continue until Q_{39} : the property persists.

If you’ve seen Robert Zemeckis’s movie Contact (1997), it should remind you of something.

A question arises: would it be true that the whole Q_{n} is first, whatever n? It must be admitted that, so far, the indices accumulate …

But when we come to n=40, nothing goes anymore !! Indeed : conjecture

    \[ Q_{40}=40^{2}+40+41=1681=41^{2}\]

Q_{40}is therefore divisible by 41. It is a composite number (ie: not first).

In the space of a few minutes, a conjecture was born, and then we saw it collapse.

2 – Good! So … a guess … what is it?

In mathematics, the term “conjecture” refers to an utterance that is believed to have a good chance of being true, but without any evidence. conjecture

Such status is – by nature – not definitive! Here are the three possible fates of a conjecture:

– We can discover, after a longer or shorter time, that it is a false statement. This is the case of the conjecture that “ n^{2}+n+41 is a prime number, regardless of the natural integer n “: This one did not resist very long to our assaults. In this case it is said that the conjecture has been refuted. A much more serious example of refutation is proposed below with the Euler conjecture .

– One can also end up establishing the validity of a conjecture: it then changes status; it becomes a theorem . A famous example is the assertion that, if the integer n is greater than or equal to 3, then there is no triplet \left(x,y,z\right) non-zero natural numbers verifying x^{n}+y^{n}=z^{n}. On a copy of Diophant’s Arithmetic which he annotated, Pierre de Fermat wrote in 1637 that he had a wonderful proof of this result, but that the margin too narrow would not contain it. For nearly 350 years, the most eminent mathematicians have contributed to the spectacular development of several branches of algebra by vainly trying to obtain a complete demonstration. Then in 1994, the British mathematician Andrew Wiles demonstrates the famous “conjecture of Taniyama-Shimura-Weil”, one of the corollaries is precisely the assertion of Fermat.

PICT

To learn more about this exciting story, I highly recommend reading a book titled “ The Fermat’s Last Theorem, “ written in 1997 by science journalist Simon Singh. This book, which is also in French translation, contains absolutely no technical passage. And reading is a pleasure from one end to the other.

– Far more rarely, mathematicians get their hands on a statement that proves undecidable: it is simply not accessible within the framework of standard mathematical theory. To be precise, it should be said: in the framework of ZFC (Zermelo-Fraenkel axiomatic + Axiom of choice), but … This may seem strange, and even implausible. However, this situation has already occurred. Schematically, we can interpret the undecidability by thinking of a game of society: it is a little as if, during a game, the players reached a position not provided for by the rules. They wonder if such action is lawful or not, without being able to decide … They can then decide to add a new rule that authorizes this action. But they can also decree an opposite rule, which will forbid it.

Thus, regularly, certain sectors of the mathematical community are agitated by eddies: after having carried out calculations, after having verified and re-verified them, a new property seems to emerge … an embryo of theorem shows maybe the tip of its nose … unless\ldots{}

So, more mathematicians are looking into the question. Each relies on his knowledge and intuition to choose what seems to be the right angle of approach: some try to refute the conjecture, others try to demonstrate.

Now, let’s review some famous conjectures: some have been established, others have been refuted, others still constitute open questions to date … And among them, some unresolved issues for centuries !!

3 – Perfect numbers, Mersenne numbers

In antiquity, Euclid (still him) was interested in perfect numbers. These are the natural numbers that are the sum of their strict divisors. For example, the integer 6 admits for divisors: 1,2,3 and 6. The strict divisors of 6 are therefore 1,2 and 3. And the sum of these is equal to 6, which makes 6 a number. perfect.

In antiquity, four perfect numbers were known: 6,28,496, and 8128conjecture

As we struggled to find a fifth, the idea that there were no others began to make its way. From there to make the link with the four elements (water, air, earth and fire), there was only one step, which was briskly crossed by some mystical spirit. Hence, it seems, the terminology of “perfect” number.

And then, towards the middle of the XVth century, a fifth perfect number was discovered (but the author of this discovery is not known): it is about 33\thinspace550\thinspace336. Then others followed …

Today, 49 perfect numbers are listed. The biggest of them is:

    \[ 2^{74207280}\left(2^{74207281}-1\right)\]

It’s an absolutely colossal whole!

Euclid understood that if 2^{p}-1 is prime (which requires p to be first, too), then the number 2^{p-1}\left(2^{p}-1\right) is perfect. It took almost two thousand years for the great Swiss mathematician Leonhard Euler (1707 – 1784) to demonstrate that, conversely, every perfect pair is of this form.

The stage is set. Now, here are two questions that remain so enigmatic:

[Qu. 1] Is there an infinity of perfect numbers? conjecture

[Qu. 2] Are there odd perfect numbers? conjecture

Nobody knows anything. This is all the more surprising as these questions are very simple to formulate! We can certainly explain to a child what a perfect number is and make these two questions perfectly intelligible. But it’s a different thing to try to answer it …

Let us add that an equivalent formulation of question 1 consists in asking if there exists an infinity of prime numbers of Mersenne. We thus call the prime numbers of the form 2^{p}-1, in tribute to Marin Mersenne (1588 – 1648), who studied them actively.

To learn more about Mersenne numbers and perfect numbers, visit the GIMPS website (acronym for: Great Internet Mersenne Prime Search).

4 – First Numbers Twins

Looking through the growing list of prime numbers, one occasionally comes across “pairs of twin primes”.

These are couples \left(p,q\right) prime numbers such as q=p+2. Here are a few :

    \[ \left(3,5\right),\quad\left(5,7\right),\quad\left(11,13\right),\quad\cdots\quad\left(10\thinspace007,10\thinspace009\right),\quad\cdots\quad\left(17\thinspace329\thinspace889,17\thinspace329\thinspace891\right),\quad\cdots\]

Is there an infinity? The question remains open to this day. Yet, significant progress has been made. The latest is the work of Chinese-American mathematician Ytang Zhang . If we note \mathcal{A}_{N} the assertion that there are infinitely many prime numbers p such as p+N is also prime, then the conjecture of the twin prime numbers is to assert that \mathcal{A}_{2} is true. conjecture

  1. Zhang managed to demonstrate in 2013 that there is an entire N<7\times10^{7} such as \mathcal{A}_{N} is true. According to the specialists, the path separating this remarkable breakthrough from a (possible) proof of \mathcal{A}_{2} is still long.

5 – The Goldbach conjecture

In a letter dated 1742 addressed to Euler, the German mathematician Christian Goldbach (1690-1764) states a conjecture whose modern formulation is as follows: “every integer peer greater than 2 is the sum of two primes”.

It is easy to see that:

    \[ 4=2+2\qquad6=3+3\qquad8=5+3\qquad10=5+5\]

(but also 10=7+3, which shows the non-uniqueness of such a decomposition)

and it continues :

    \[ 12=7+5\qquad14=7+7\qquad16=13+3\qquad etc\cdots\]

With powerful computers, all even integers up to very large values <Peringatan LyX: Karakter tidak bisa di encode ‘​’><Peringatan LyX: Karakter tidak bisa di encode ‘​’>(about 4 billion billion!) Have been systematically tested, without any counterexample being found! Of course, this proves nothing (think of the integers of the form n^{2}+n+41 mentioned at the beginning of the article). conjecture

Again, remarkable progress has been made. For example, mathematician Chen Jingrun has demonstrated that any sufficiently large even number can be written either in the form p+q with p and q first, in the form p+qr with p,q and r first. The proof of this result was published in 1966 and significantly simplified thereafter. One can think, reading the statement of this theorem that the final outcome is not very far, but again, in the opinion of specialists, it will take a lot of work to elucidate the conjecture of Goldbach. conjecture

6 – Fermat numbers

The n-th Fermat number is defined by F_{n}=2^{2^{n}}+1.

Specify that 2^{2^{n}} denotes the integer 2 raised to power 2^{n} (and not : 2^{2} raised to power n … otherwise we would have written 4^{n}).

It is easy to calculate:

    \[ F_{0}=3,\quad F_{1}=5,\quad F_{2}=17,\quad F_{3}=257,\quad F_{4}=65\thinspace537\]

These five numbers are all first.

Pierre de Fermat (1601-1665) conjectured that F\_\{n\} is first for everything n\in\mathbb{N}.

A century later, Euler belied this claim by observing that:

    \[ F_{5}=4\thinspace294\thinspace967\thinspace297=641\times6\thinspace700\thinspace417\]

Today, it is established that some Fermat numbers are composed (ie: not first), but it is still not known if there are any that are prime, apart from the five mentioned above. We conjecture that F_{n} is prime for no integer n>4 … but to date, no proof (:

If you can prove that, feel free to let it know: you will instantly become a global celebrity.

Before closing this overview, let us quote two more examples of famous conjectures.

7 – The Euler Conjecture

The sum of two perfect squares can sometimes give a perfect square. For example :

    \[ 3^{2}+4^{2}=5^{2}\qquad\text{or}\qquad12^{2}+5^{2}=13^{2}\]

This is known since ancient times: triplets of natural numbers \left(x,y,z\right) checking x^{2}+y^{2}=z^{2} are called “Pythagorean triplets”. Describing them all, using an explicit formula , is a classic exercise in arithmetic.

Can a sum of two cubes give a cube? Fermat’s assertion says no. For a sum of several cubes to be equal to a cube, it must therefore have at least three terms. This is achieved with three terms, as shown in the following example:

    \[ 3^{3}+4^{3}+5^{3}=6^{3}\]

Then the question arises: conjecture

If the sum of n powers k it’s still a power k-th, is it necessary that we have that n\geqslant k?

Euler conjectured that yes. But in 1966, LJ Lander and TR Parkin exhibited a first counterexample:

    \[ 27^{5}+84^{5}+110^{5}+133^{5}=144^{5}\]

which puts forward a fifth power that can be broken down into the sum of four such powers! Euler’s claim was contradicted for k=5.

Then in 1986, N. Elkies proposed:

    \[ 2\thinspace682\thinspace440^{4}+15\thinspace365\thinspace639^{4}+18\thinspace796\thinspace760^{4}=20\thinspace615\thinspace673^{4}\]

which constituted a new attack against this conjecture, for k=4 this time. The smallest counter-example for the exhibitor 4 was found two years later by R. Frye:

    \[ 95\thinspace800^{4}+217\thinspace519^{4}+414\thinspace560^{4}=422\thinspace481^{4}\]

To this day, the mystery remains intact for the exhibitors superior or equal to 6.

8 – The Catalan Conjecture

A natural whole number is called a “perfect power” when it is of the form a^{n}, or a and n are integers such as a\geqslant1 and n\geqslant2.

Thus, the numbers 16 and 125 are perfect powers, since:

    \[ 16=2^{4}\qquad125=5^{3}\]

16 is therefore a “perfect square”, while 125 is a “perfect cube”.

The number 3\thinspace404\thinspace825\thinspace447 is also a perfect power since:

    \[ 3\thinspace404\thinspace825\thinspace447=23^{7}\]

but a little less obvious.

The Franco-Belgian mathematician Eugène Catalan (1814 – 1894) publishes in 1844, in the Journal de Crelle , the following note:

“I beg you, Sir, to be good enough to state in your collection the following theorem, which I believe to be true, although I have not yet succeeded in demonstrating it completely. Others may be happier: two consecutive integers, other than 8 and 9, can not be exact powers; in other words: the equation x^{m}-y^{n}=1, in which the unknowns are whole and positive, admits only one solution. “ To make short: Catalan conjectures that 8 and 9 are the only two consecutive perfect powers.

Let’s say a few words about the origin of the problem:

Philippe de Vitry (1302 – 1357), Bishop of Meaux, was passionate about theory and musical composition. It is perhaps due to the treatise Ars Nova Musicæ, published around 1320. He made the following observation concerning the intervals of notes: the octave, the fifth, the fourth and the tone respectively correspond to reports \frac{1}{2}, \frac{2}{3}, \frac{3}{4} and \frac{8}{9}. Today, we would talk about frequency ratios, but at that time we probably had to refer to string length ratios. Surprising of his values, he wondered if they were the only pairs of consecutive “harmonic numbers” (a harmonic number being of the form 2^{a}3^{b}).

Levi Ben Gerson (1288 – 1344) was a scholar living in the south of France: he was a philosopher, astronomer, commentator on the Bible and a mathematician. Philippe de Vitry asked him to write a mathematical proof that would confirm (or not) this conjecture. This is what Ben Gerson did: he wrote a demonstration in 1343, a year before his death. The original (in Hebrew) has been lost, but a Latin translation (written for Philippe de Vitry) has reached us. We can also extract today a small arithmetic exercise , posable in terminal S (specialty maths).

Later, in 1738, Euler proves that the couple \left(3,2\right) is the only solution of the Diophantine equation x^{2}=y^{3}+1.

In 1850, (six years after the note published by Catalan), Victor Amédée Lebesgue (1791 – 1875) demonstrates that if p is prime, then the equation x^{p}=y^{2}+1 has no solution. He concludes his article with these words:

“The other cases of the equation x^{m}=y^{n}+1 seem to be more difficult. I have not been able to know so far what Mr. Catalan has found on this subject. “

In 1885, Catalan publishes his “scientific testament”: a book entitled “Mathematical Mixes” in which he lists various results he has achieved in his life as a mathematician, as well as various avenues of research that have not succeeded. Regarding the conjecture that bears his name, he writes:

“After almost a year’s loss in search of a flying show, I gave up this tiring search.”

The conjecture of Catalan finally became a theorem, but after a little more than a century and a half of waiting!

It was indeed in 2002 that the Romanian mathematician Preda Mihailescu managed to establish this result, which was a tour de force. Here again, the contrast between the simplicity of the utterance and the immense difficulty of the proof is striking.

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