I saw on twitter this morning a long time ago [it was a long time ago when I wrote this post but didn’t publish it] (from Jeff Atwood, of Coding Horror blog and Stack Overflow fame) the following elegantly and concisely stated counter-example that would – if true – disprove perhaps the most famous of mathematical theorems, Fermat’s Last Theorem (FLT):
1782^12 + 1841^12 = 1922^12
Wow! A counter-example for FLT. A theorem I’ve known about since I was a kid. One counter-example is all it takes to disprove the whole deal.
Fermat’s theorem states that the equation an + bn = cn has no solutions for integer n > 2, and integers a, b, and c not equal to zero. For n = 2 we have many solutions (Pythagorean triples), but none for n > 2. Nor should we, according to English mathematician Andrew Wiles, who proved FLT in 1995.
Until now. Or do we? The equation Jeff posted is a little awkward to validate since most calculators cannot handle numbers this size at full precision. They appear equal with a normal calculator – due to precision limits (round-off errors). Same problem with Excel.
So, since I’ve recently started playing with F#, I put together a trivial F# program (included below) to show the math at full precision, with the following results:
1782^12 + 1841^12 = 2541210258614589176288669958142428526657
1922^12 = 2541210259314801410819278649643651567616
which differ by
So Fermat is safe. Saved by F#. 🙂 But don’t feel bad if you fell for it – just be glad you knew what it meant. Bonus if you noticed it on The Simpsons or Futurama.
Just watched the The Fountainhead movie from 1946 (yes, from netflix).
Howard Rourke hacking Open Source
Here is the plot summary, brought up-to-date:
- Open Source is represented by the protagonist, a brilliant architect named Howard Rourke. Rourke is idealistic, does his own thing, is uncompromising, and is not driven by money or recognition – and certainly not by Big Business.
- Big Business is represented by newspaper magnate Gail Wynand. Wynand wields substantial influence and is in perpetual pursuit of any means to incite the populace – an energized populace buys more product.
- Consultants and Certified Vendor X Developers and Vendor Partners are represented by architect Peter Keating. Keating goes with the flow, producing whatever the powers that be say is desirable. At one point, he mentions to Ms. Francon he’s polling folks on what they think of Rourke’s latest building to which she responds (with some disdain) “why, so you’ll know what you think of it?”
- Talent != influence. Keating’s influence is limited to those who recognize his greatness. Most only recognize as great what they are told to recognize as great.
- Passion can be directed constructively (Rourke pours his love into his life’s work) or destructively (Wynand devotes his career to controlling the masses through his newspaper).
The movie is based a book of the same title. The author, Ayn Rand, became well known for her Objectivism philosophy of life, exemplified in the movie by Gary Cooper who played the lead character, Howard Roark. [I wonder what Richard Stallman thinks of the book?]
I wonder how many professional software developers identify more with Howard Rourke or Peter Keating? And which is more desirable?
Any my clean analogies fall apart when one considers the combinations of Big Business and Open Source. Microsoft just announced CodePlex.org and the CodePlex Foundation “to enable the exchange of code and understanding among software companies and open source communities.”
A dirty little secret of Eclipse, Linux, Apache and other high-profile projects is that they also have professional, full-time staff – sponsored by Big Business (like IBM) – since the success of these endeavors is strategic for their business.
Maybe Open Source isn’t as pure as the romantic notion of developers from around the world contributing since it was a nice thing to do. The world-wide altruistic contributions may still be there in some cases, just supplemented by Big Business. Which is okay with me, though might not be with Howard Rourke.