The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet."
The English translation is: "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
Here at SAS, we don’t take challenges lightly. After a short but intensive brainstorming, we came up with a creative and powerful SAS code that effectively proves this long-standing theorem. And it is so simple and short that not only can it be written on the margins of this blog, it can be tweeted!
Drum roll, please!
Here is the SAS code:
data _null_; do n=3 by 1; do a=1 by 1; do b=1 by 1; do c=1 by 1; e = a**n + b**n = c**n; if e then stop; end; end; end; end; run;
Or written compactly, without unnecessary spaces:
data _null_;do n=3 by 1;do a=1 by 1;do b=1 by 1;do c=1 by 1;e=a**n+b**n=c**n;if e then stop;end;end;end;end;run;
which is exactly 112 character long – well below the Twitter 140-character threshold.
Don’t be fooled by the utter simplicity and seeming unfeasibility of this code. For the naysayers, let me clarify that we run this code in a distributed multi-threaded environment where each do-loop runs as a separate thread.
We also use some creative coding techniques:
1. Do-loop with just two options, count= and by=, but without the to= option (e.g. do c=1 by 1;). It is a valid syntax in SAS and serves the purpose of creating infinite loops when they are necessary (like in this case). You can easily test it by running the following SAS code snippet:
data _null_; start = datetime(); do i=1 by 1; if intck('sec',start,datetime()) ge 20 then leave; end; run;
The if-statement here is added solely for the purpose of specifying a wait time (e.g. 20) sufficient for persuading you in the loop’s infiniteness. Skeptics may increase this number to their comfort level or even remove (or comment out) the if-statement and enjoy the unconstrained eternity.
2. Expression with two “=” signs in it (e.g. e = a**n + b**n = c**n;) Again, this is a perfectly valid expression in SAS and serves the purpose of assigning a variable the value of 0 or 1 resulting from a logical comparison operation. This expression can be rewritten as
e = a**n + b**n eq c**n;
or even more explicitly as
e = (a**n + b**n eq c**n);
As long as the code runs, the theorem is considered proven. If it stops, then the theorem is false.
You can try running this code on your hardware, at your own risk, of course.
We have a dedicated 128-processor UNIX server powered by an on-campus solar farm that has been autonomously running the above code for 40 years now, and there was not a single instance when it stopped running. Except pausing for the scheduled maintenance and equipment replacements.
During the course of this historic experience, we have accumulated an unprecedented amount of big data (all in-memory), converted it into event stream processing, and become a leader in data mining and business analytics.
This leads us to the following scientific conclusion: whether you are a pure mathematician or an empiricist, you can rest assured that Fermat's Last Theorem has been proven with a probability asymptotic to 1 beyond a reasonable doubt.
Have a happy 91-st day of the year 2017!