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Message from discussion The story behind a formula for Pi

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More options Jun 24 2003, 2:14 am
Newsgroups: sci.math, sci.math.symbolic
From: plou...@math.uqam.ca (Simon Plouffe)
Date: 23 Jun 2003 23:14:32 -0700
Local: Tues, Jun 24 2003 2:14 am
Subject: The story behind a formula for Pi
This note explains the story of the so-called Bailey-Borwein-Plouffe
algorithm
and formula.

The story began many years ago in 1974 when I wanted to find
a formula for the n'th digit of Pi. I was studying rational and
irrational
numbers. With my calculator I was computing inverses of primes and
could
easily find a way to compute those inverses in base 10 to many digits
using
congruences and rapid exponentiation. Since it appeared impossible to
do
the same for Pi, I wanted then to find a simple formula f(n) that
could compute
the n'th digit of Pi. I had that idea for 20 years.

Since the computation of Pi looks
more complicated than the number E , i.e. exp(1), I studied a way to
compute
that number instead. At that time (around 1983), I had a simple Basic
program
that used a spigot algorithm to compute E, as expected that algorithm
worked but
of course but was taking an increasing amount of memory. My question
was :
why can't we do it for E or Pi or any irrational numbers like sqrt(2).

It was during the year 1994 that I began to compute arctan series but
I did not
realized that this meant a lot. I was able to use an algorithm to
compute arctan
of 1/5 with fast exponentiation without realizing that it could
compute arctan(1/5)
in base 5 very fast since the rapid exponentiation was natural in that
base.

Later in 1995, around august 7 of that year I suddenly realized that
log(2) was
fast computable in base 2. Since I had a bit of experience with spigot
algorithms
and also my little Basic program to compute arctan, it was not
the algorithm to log(2).
In the next few days I made my first program : A program to compute
log(9/10)
in base 10 using a very small amount of memory and very fast. The
432 characters long.

That discovery was a shock to me. I realized that I had found it yes
but it was not
new to me since I could do arctan(1/5) easily too but it took me 2
years to realize it.

This is where I began to use Pari-Gp, that program could find an
integer relation
among real numbers (up to a certain precision), very fast.

During my stay at Bordeaux University in 1992-1993 I perfected that
that could interface Pari-Gp and Maple. That little Unix script had an
enormous
advantage of flexibility because I could set up a series of real
numbers to test among 1
unknown. At that time I was beginning to find new results, the
programs were able to
find identities.

That program was the one that found the formula for Pi in hexadecimal
(or binary).
I also used another one : PSLQ. It was a good program but a bit
cumbursome to use
since it is written in fortran. Nevertheless I made an interface to
Maple too.
Pari-Gp was by far easier to use and faster for small cases (up to 10
real numbers at
the time with 100 digits precision was enough for those kind of
problems).

This is where I made the biggest mistake in my life : To accept the
collaboration
of Peter Borwein and David H. Bailey as co-founders of that algorithm
and formula
when they have found nothing at all. David Bailey was not even close
to me when
I found the formula. He was added to the group 2 months after the
discovery.

I was naively thinking that I could negociate a job as professor at
Simon Fraser
University, which failed. I am very poor at negociations.
I remember that day when the Globe & Mail newspaper article went out
in October
1995. I was at Jon borwein's house and he had a copy of the newspaper
in hand.
This is where I asked him to become a professor at SFU. He simply
replied right
away < don't even think about it >. I thought, this is the best chance
I will ever have
to become a professor there, since it failed, I decided that I had to
leave that place.

I was very frustrated at that time, in late 1995 after the discovery.
I realized that
many small details where terribly wrong. They were getting a lot of
credit for the
discovery and I had the impression of not getting anything in return.
My strategy
failed. One of those details was the article of the Globe and Mail, I
: why did they putted the photo of you and your brother on the article
has nothing to do with this!.  He simply replied that the Public
Relations at the
Later that year, I was invited to a ceremony in Vancouver for the CUFA
(faculty of
the year Award).
This is a prize with plaque and mention that those 2 brothers received
for the
discovery of the formula. They simply mentioned my name at the
ceremony and
I received nothing at all. They made a toast to the queen of England,
I did not
stand up.

In late 1995, there was that Canadian Math Soc. congress in Vancouver,
I was not
invited to talk about the discovery. There was even a guy (Stan Wagon)
that said
to me, I don't know if you have anything to do with this but in all
case, this is
good for you isn't ?

Then in 1996, I realized that if I get up at night to hate them it is
it means that I have to leave that place (Simon Fraser university).
I was convinced I had no future at all with those 2 guys around.
I was making serious plans to leave.

The story of the formula (my formula), was not the only one. The same
thing happened
with the ISC (the Inverse Symbolic Calculator). The story is even more
ridiculous.
I opened the site with my constants in July 1995 and it was an
immediate success.
tables and all
of the Unix programs to run it. The precious help I had was from Adam
Van Tuyl, a
graduate student, he made most of the code behind the web pages, later
Paul Irvine

At that time the local administrator of the lab. tried to convince me
to stay even to pay me
for maintaining the ISC, I refused. I wanted to leave with what I had
: my tables of
real numbers and sequences I worked for years (since 1986). This is
why I opened the
Plouffe Inverter with my name in 1998, to keep what was mine.
When I realized that I was about to loose the paternity of the ISC, I
left in march 1997.
I went to Champaign Illinois to work for Wolfram and Mathematica.
(this time it took me less time), that one was worst than the 2
brothers combined. I simply
left as soon as I could, 5 months later.

Peter Borwein wanted very much that I do a Ph. D. on the ISC but he
wanted also to
publish (with his name of course) an article before I deposit the
thesis. Again the
same story was going on, these 2 guys are so greedy I can't believe
it. The behavior they
had with me was not exclusive, especially Peter Borwein he was the
same with most of
his students, especially the good ones, sucking the maximum. Jon is
the same but he
has more talent in politics (more money too). He is good but has a
tendency to site
himself a lot. He thinks that if he had the idea of the sum of 2
numbers at one point
in his life then all formulas in mathematics are his own discovery.

About David H. Bailey. He came after the discovery of the formula and
my small basic
program , I had also a fortran version. This is where Peter Borwein
him as a collaborator to the discovery since he contributed to it (as
he said), this is my
second big mistake. Of course he accepted to co-write the article, who
wouldn't ?!
David H. Bailey (and Ferguson) are the authors of the PSLQ program.
That program is
the <american> version of the Pari-Gp program. I used it a little it
the discovery was pari-Gp and Maple interface program I had. So
actually, that person
has nothing to do with the discovery of that algorithm and very little
to do with the
finding of the formula. The mistake was mine. Saying that Bailey found
the formula
is like saying that the formula was found by the Maple and Basic
program.

I tried very hard to correct the situation avoiding the subject of the
actual discovery
of the algorithm and the formula, I made an article in 1996 for the
base 10. I thought
naively again that this would re-establish the situation, it did not.
I almost accepted
to do a film at one point in 1999 when a certain guy from England that
wanted to make a
movie on Pi and the discovery of the formula. he asked me if I would
accept to talk
about my <differents> with the Borweins. I did not wanted to go in
that direction,
I should had. There was that book of Jean-Paul Delahaye (le fascinant
nombre pi)
that mentioned the Plouffe algorithm and formula because I told him
part of the
story. In some way I was afraid of revealing that enormous story.

Why was I so naive ?
I had a previous collaboration with Neil Sloane and the Encyclopedia
of Integer Sequences
and the web site, this was really a big success and Neil is the person
I respect the most
in mathematics so this is why I thought (wrongly ) that my
collaboration with the
Borweins had to go well, a big mistake.

Why do I write this ?
To tell the truth and also the arrogance of those people makes me
sick.

Will I gain something from this ? I don't care, I have nothing to
loose.

Simon Plouffe
Montréal, le 22 juin 2003.