The community of mathematicians is in no hurry to accept the proposed proof
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Has the Riemann hypothesis been proven?
A Purdue University mathematician has submitted a proposal to prove the Riemann hypothesis, one of the great unsolved problems of our time. The proof has not yet been tested
Alluvial container, Galileo
Mathematician Louis de Bernage, who had previously made several unsuccessful attempts to prove the Riemann hypothesis, made the proposed proof public before it was published in a scientific journal and before it was checked by others. The rapid and widespread publication came to declare its rights to the proof, in case another was found to offer a similar version.
A reward for the prover and not the disprover
This is due to a prize of one million dollars, which will be given by the Mathematical Institute of Cambridge, Massachusetts, to whoever proves this conjecture. It is interesting to note that if someone disproves the Riemann hypothesis they will not win the prize. This may be because a proof is a pure mathematical work, while a refutation can also be obtained by a computer, which will find a single contradictory example.
The Riemann hypothesis, which was first presented in 1859, was never proven, not even by Bernard Riemann himself who tried to do so for the following seven years, until his death in 1866. This is a hypothesis concerning, among other things, prime numbers and how they are distributed among the other numbers. The hypothesis refers to a function known as Riemann's zeta function.
Zeta function
This function is important in mathematics and physics, not all of its properties have been fully explored. The zeros of the function, i.e. the values that can be placed in it so that the result is zero (i.e.: the "solutions" of the function), are divided into two types: "trivial", which are all even negative numbers (such as -2, -4, -6), and non-trivial , which are complex numbers.
A complex number is a sum of a "normal" number, i.e. real, and an imaginary number. An imaginary number is a root of a negative number. The root of 1- is denoted by the letter i. So, for example, the root of 4- is 2i. A number like 2.5+3i is a complex number, and it is customary to present it as a point on the "composite plane", then the horizontal axis represents the real part of the number, and the vertical axis represents the imaginary part.
softened numbers
The Riemann conjecture states that in a certain presentation of the Riemann zeta function, the resulting solutions are all complex numbers with a real part equal to 1/2, and any imaginary parts. If we present the solutions on the composite plane, it will appear that they are all on a vertical line, whose value is on the real axis ½. It seems that Riemann did not reach this conclusion by the power of pure intuition, but by careful manual calculation of some of the function's solutions.
What about prime numbers? The Riemann hypothesis is closely related to another mathematical claim, concerning prime numbers. This is a claim that stems from the Riemann hypothesis (if proven), but it has already been proven in the past in other ways. The prime number theorem refers to the question of how many prime numbers exist in a certain range. For example, how many prime numbers are there up to the number 100? twenty-five. And how many up to the number 1000? One hundred sixty-eight. And how much to 10 to the power of 10?
average density
It turns out that a good estimate can be obtained using a function that is an integral over a logarithm. Its meaning is a kind of "average density" of the initial numbers. It can be said that according to the theorem, the probability that any number n is a prime number is given by 1/log(n). There are still many unsolved problems related to prime numbers. The motivation for solving them is partly purely intellectual and partly applied, since prime numbers are particularly useful in encryption. Among the unresolved questions:
How many "pairs" of prime numbers, the difference between which is 2, exist? Pairs are, for example, 3 and 5, or 29 and 31. There may be infinitely many of them.
For every number n, is there a prime number in the range between n2 and (n+1)2?
The study of prime numbers is also developing through numerical means (using a computer). By such means the largest prime number currently known to mankind was recently found, albeit 7 million digits. The number, published in June of this year, can be presented as: 2 to the power of 24,036,583 minus one.
A mathematician claims to have proof of the Riemann hypothesis
Dikla Oren, 13/6/2004
A French mathematician claims to have solved a particularly difficult problem, for which a million dollar prize is being offered. However, other mathematicians are not in a hurry to accept his proof.
On Tuesday, Louis de Branges de Bourcia, a professor of mathematics at Purdue University in Indiana, issued a press release in which he claims to have proved the Riemann hypothesis.
This proof is perhaps the most desired proof in mathematics today. If true, the Riemann hypothesis states that prime numbers, numbers divisible only by themselves and by one, are distributed completely randomly along the number line. If the hypothesis is incorrect, mathematicians may be able to predict the location of prime numbers.
Mathematicians have been struggling to determine whether the Riemann hypothesis is correct for more than 150 years. De Branche previously claimed to have solved the problem, but other mathematicians found flaws in his work.
"For the last 15 years, he's been coming back and claiming he has the proof and publishing papers," says Jeffrey Lagarias, a mathematician at AT&T Laboratories in New Jersey, who follows de Branche's work.
zeta functions
The last article is a 124-page document entitled "Riemann Zeta Functions" and appears on De Branche's website. The paper has not yet been submitted to a scientific journal, and mathematicians New Scientist spoke to were not at all convinced that the paper contained a proof.
"My guess is that this is another failure in a long line of failures," says Andrew Odlyzko, a mathematician at the University of Minnesota who studies the Riemann hypothesis. LaGrias says: "I don't think he has proof right now."
Hall de Branche has had success in the past. Twenty years ago he had to face criticism when he claimed to have solved another age-old mathematical problem. In that case, he turned out to be right after all.
"De Branche always thinks he has proof. Maybe he's right, maybe not," says Harry Diem from the Weizmann Institute in Rehovot, Israel. The problem is, Diem says, that the frequency with which he claims proof "has created a reluctance among people to spend time checking a new proof."
To claim the $XNUMX million prize, offered by the Clay Institute of Mathematics in Cambridge, Massachusetts, de Branche must first publish his paper in a journal, and the paper must survive two years of rigorous scrutiny by the mathematical community. De Branche's plans to use the winnings to restore an ancient castle in France and turn it into a mathematics institute, therefore, will remain on hold for the time being.
Translation: Dikla Oren
The article in New Scientist
The math genius
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