### The story behind a formula for Pi

Perm url with updates: http://xahlee.org/math/Simon_Plouffe_pi_formula.html

# 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.