A proven mathematical guess after 80 years

Denominators are shown on this graph. They are related when they have at least one common prime factor, meaning a prime number that can divide them into an integer. For example, 25 and 10 can be divided by 5. Unrelated denominators, such as 11 and 20, do not share any prime factors. It is the analysis of the pattern of such a graph that has enabled mathematicians to demonstrate that even when the denominators chosen share many prime factors, the Dauphin Schaeffer conjecture still holds. Adapted from Quanta Magazine

We do not prove every day a mathematical conjecture, a statement that is verified by examples, but not proven beyond a reasonable doubt. However, this is what Dimitris Cocoulopoulos, a mathematician at the University of Montreal, achieved by validating the Dauphin Schaeffer conjecture formulated 80 years ago … a feat praised by mathematicians around the world!

This conjecture revolves around the question: Is it possible to approximate, using a simple fraction, a number like Pi with a certain precision? Pi is part of the irrational number family. It is obtained by dividing the circumference of a circle by its diameter. And it’s quite elusive, because it contains an infinite number of decimals: 3.141.592.653.589 … but the vast majority of the numbers that are present are illogical. The mathematician learned to reconcile these numbers, which cannot be expressed as a fraction of two integers.

On the other hand, he says, “we can approximate it by a fraction.” For example, 7/22 gives 3,142 857 14 …, which corresponds to Pi to its second decimal place (hence the numbers in bold). In fact, the difference between this approximation and the true value of Pi is 0.001 … which is fairly acceptable as a margin of error for such a simple fraction.

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If you try with a fraction with a larger denominator (the divisor), you may get a smaller margin of error. Thus, 355/113 gives 3,141 592 92 …, which approaches Pi to its sixth decimal place. Margin of error, here 0.000 000 26…, very respectable! Engineers often use this type of fraction instead of Pi in their calculations.

In the sequence of fractions, the accuracy of the approximation of Pi increases with the increase in the denominator:

Mathematicians have long bothered approximating irrational numbers (such as Pi) with integer fractions. In 1941, American mathematicians Richard Dauphin and Albert Schaeffer, in the conjecture that kept their names, assumed that when one chooses a series of numbers as a denominator (for example, prime numbers, even numbers, or all numbers ending in 7. …) and we choose a margin of error It is not crossed, two situations occur: it is possible to approximate almost all irrational numbers or after that we cannot approximate any of them! “There is no mediating position,” said Demetris Kokolopoulos. There is no case where we can approximate 30 or 50%. All-or-nothing. “

Although a handful of theorists validated small bits of the guesswork, they resisted all attempts. Demetris Cocoulopoulos and colleague James Maynard, from the University of Oxford in England, used an original approach to come up with a 55-page demonstration, published in July 2020 in Annals of Mathematics.

“Everyone surprised us a little by attacking the problem with graph theory. Very briefly, it is about putting points that represent denominators of a series on a graph and connecting them together when they have a lot of common factors. When there are too many connections, approximation is impossible. Ultimately, they will not allow rounding of nearly any irrational number.

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Jean-Marie de Konink, a famous mathematician from Laval University who did not participate in the study, expresses with a smile in his voice his admiration for their intuition: “These two mathematicians are geniuses with great potential. It is not for nothing that their demonstration appeared in Annals of Mathematics. It is the standard in our field and everyone aspires to post there during their careers. They are barely 35 years old! What are you jealous of! “

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