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!):
http://www.lacim.uqam.ca/plouffe/Chudnovsky.html
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 *10*^{79} atoms that there is in the universe representing
the set of theoretically exploitable resources, we will not be able to calculate
more than *10*^{77} 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.
Trials
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:
*(6log(6)-3log(3*6403320))n=14,18n*
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 !
back to home page |