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# Math Politics: Simon Plouffe and The nth Digit Formula of π

Xah Lee, 2010-12-27

Discovered a notable math politics.

There's a fomula, that can compute the nth digit of pi directly. The formula is named
Bailey–Borwein–Plouffe formula.
Presumably, it is discovered by 3 persons
David H Bailey, Peter Borwein, and Simon Plouffe, but actually might be just a single person Simon Plouffe.
This is a typical politics in math or other science community.
Here's what Simon has to say about it.

## The story behind a formula for Pi

From: plou...@math.uqam.ca (Simon Plouffe)
Date: Jun 23 2003, 10:14 pm
Subject: The story behind a formula for Pi
To: sci.math, sci.math.symbolic

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 difficult to adapt 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 program had 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
program I had 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
asked Peter Borwein : why did they putted the photo of you and your
brother on the article ? Your brother has nothing to do with
this!. He simply replied that the Public Relations at the University
made a mistake. 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 a very bad sign, 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. The 2
Borweins had nothing to do with that thing, I had made the 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 made some additional code.

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 suggested to add 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 is true, but what made 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.

✻ ✻ ✻

The original post can be seen here: http://groups.google.com/group/sci.math.symbolic/msg/5b7e62ce42ae0cb6

I do not know Simon, but i tend to believe him. Wikipedia seems to indicate so too.

I happened to have exchanged few emails with Simon around later 1990s, and he has a paper on Cardioid on my site. See the bottom of the page: Cardioid.

Simon has a home page at: http://lacim.uqam.ca/~plouffe/.

PS Thanks to meowcat for mentioning the formula to me.

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