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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013

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Grégory and David Chudnovsky

Important formula

Slices of life

The Chudnovsky brothers have an extraordinary personal history! Can this too short page and some others, honour these great passionates of Pi...
They are Ukrainian, they lives from 1977 in New York near Columbia's university and...
But let us begin with the beginning, otherwise, how could we grasp this extraordinary route what is theirs?
David, the great brother of Gregory, discovered mathematics rather young at Kiev by devouring a book What is it that mathematics? By R. Courant and H. Robbins. This book was very popular in spite of its illegality in Russia and China of then...
David decided after this reading to become a mathematician and his younger brother quickly followed close David's heels. Gregory began to publish very fast, from 16 years old in Soviet mathematics (Some results in theory of infinitely long expressions), encouraged by his brother who felt his genius. And for cause, he resolved Hilbert's tenth problem at 17 years old ( 1970 ) by proving the undecidability of diophantian equations, and this just a little after Matyasevitch, another young mathematician who recognized nevertheless that Gregory's method was better than his!
The two brothers study at the university of state of Kiev and taken in for their thesis at the academy of the sciences of Ukraine.
From the middle of the 70's, they published together. But there was nevertheless a big problem... Gregory is indeed affected of a rather rare disease of muscles degeneration which obliges him to lie down most of the time and to frequently go at the hopital... So his health is very fragile...
That is why in 1976, the parents of Gregory, Volf and Malke, asked the authorities to let them emigrate to take care of him. Very bad idea because the KGB began to harass them, Volf lost his work, and David and Malka were attacked afterward...
Edwin Hewitt, a mathematician of the university of Seattle collaborated with Gregory in 1976 and, by hearing of the problems of his family, persuaded an influential senator to put pressure on the Soviets. A French parliamentary delegation visited secretly Chudnovsky, and two months later, at the end of 77, the Russian government gave up and let the family leave. David got married with one of the women of the delegation (Very pretty is seems...) and with his family, he emigrated towards France, then in the United States. Chudnovsky settled down in New York, near Columbia university.
Nevertheless, Gregory was not able to accept an appointment because of his infirmity and David did not want to work without his brother... Result, the two brothers have no real job and are only members (senior research) of the university. So they are a little isolated, besides David is bad-tempered, it seems... The American mathematical community is hardly annoyed with these two brothers who have no precise place, with all the financial consequences that we can imagine...
So Chudnovsky works from their part, and it is not lost of time! Because David is a very good mathematician and Gregory ( 46 years) is considered simply as one of the best, having received number of awards and mastering all the branches of mathematics, as one century ago a certain... Hilbert !
See comparison !
And besides, these two boys work on the theory of the numbers, their speciality, but in additiont, on Pi, that adds more of the interest to any student in mathematics could already show interest to these two figures of mathematics.
Chudnovsky lives in a small apartment where pile up papers and computers in an incredible disorder... They moreover built themselves several "supercomputers" as the mythical m - zero, profiting by personal architecture which confers it a power upper than some Cray. It is thanks to it that they calculated billions of decimals of Pi (a good means to test it!) and they can not switch it, otherwise it would not restart so much the architecture is complex and the constituents very fragile. It contained, in 1993 , 16 microprocessors in parallel.
So the apartment of Chudnovsky is warmed with microprocessors, reaching an unbearable heat in summer !
Let us note to finish that Chudnovsky have no personal site and that it is not so easy to obtain information. Nevertheless, a very complete and very long article will be found on Chudnovsky on the following page (accommodated by Simon Plouffe!):
I have no news about them since 1994 and their last record...

About Pi

Their need to calculate decimal of Pi is connected to their conviction that there is some organization in these decimal. Because Pi is perfectly determined. And as we know nothing about properties about the transcendants numbers... We did not even prove that Pi was normal, that is to say that every digit appears once on ten, every couple once on hundred, etc.... But as with the 1079 atoms that there is in the universe representing the set of theoretically exploitable resources, we will not be able to calculate more than 1077 Decimal of Pi, it is desirable that our favourite constant shows some signs before this ultimate limit! Otherwise, " it would be terrifying! " As says Gregory...
The formula below looks like a Ramanujan, series, so with linear convergence, but only containing in the sum rationals, that improves the speed of calculation. Let us add that the two brothers program carefully their algorithms and we explaine there the fact that the champion Kanada and his big means was exceeded in 1989 and 1994 by Chudnovsky bothers with their simple supercomputer in the race of decimals! (See historic records)
Naturally, as concerning Ramanujan and Borwein, I have no exact demonstration of this formula, so refer to these two sites for explanations on this type of sum and their law of general forming.


Well , all this is not going to prevent us from trying the formula!

A priori, if Stirling's equivalent is applied to the term of the sum, we obtain :

So by applying -log to obtain the number of decimal by iteration:


Not bad!
But let us verify all this (look out, the equivalence of Stirling is asymptotic, we can not find 14 decimal by iteration for n weak):
Obviously, my computer is a little pitiful, but in few months, it will be better, We promise...

n=0 13 decimals exact
n=1 27 decimals exact
n=2 41 decimals exact
n=3 55 decimals exact
n=4 69 decimals exact
n=5 84 decimals exact
n=6 98 decimals exact

Well, such linearity is rather remarkable !

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