Have a Little Fun with Fermat’s Last Theorem

The legendary prize money for solving Fermat’s Last Theorem has already been handed over to Andrew Wiles, who used esoteric mathematics that had not been developed in Pierre de Fermat’s time to prove that the old French man really was on to something when he made a little margin note in his copy of a book on Diophantine Equations.

A Diophantine Equation is anything in the format of Xn + Yn = Zn.   We typically evaluate these equations in (positive) integer values and use them to analyze triangles (although they can be used in other ways).

Fermat said that he had discovered what he called a marvelous proof that you could not raise two integers to any power greater than 2 and combine them to form an integer raised to the same power.  Unfortunately, he didn’t have room in the margin to include his proof.

Hence, for about 300 years everyone with a pencil and paper (or computer) tried their best to solve the puzzle: was Fermat correct?

Many great mathematicians tried and failed, and far more not-so-great mathematicians (including me) gave it their best shots.  The academic authorities charged with evaluating all the proposed proofs became very adept at spotting the first flaws and rejecting the arguments quickly.

The tricky thing about Fermat’s theorem is that if you use good old Arithmetic and a little Algebra and maybe even some Calculus you can prove it is true for many different conditions but not all of them.

And so one of the great mysteries of all human existence remains unsolved for now: did Fermat really have a proof of his conjecture or did he later realize he had overlooked something?  Whole careers have been spent trying to argue for proofs of points of view on this question, which has been deemed nearly as intriguing as the conjecture itself.

Well, I can’t help you find the answer to the question but I have often tried to visualize Fermat’s rule in geometric shapes.  One of my earliest attempts consisted of imagining square sheets of paper “standing” (so to speak) in a row.  You could have, say, 100 of these sheets.  Let’s say they all have a dimension of 100×100.

That is essentially a cube (although it looks nothing like a geometrical cube).

Suppose you take another series of square sheets of paper, say 64 of them, and give them dimensions of 64×64.  That’s another cube.

Finally, imagine a smaller series of 36 sheets of paper, each of dimension 36×36.  You have a third cube.

When you add up the area of all these sheets of paper, you see that the 100x100x100 series covers far more area than the other two series.  Hence, there is no way that those two smaller cubes can equal the largest cube.

One of my friends was so impressed with this illustration she called up a mathematician friend of hers (a guy with a real Ph.D. in mathematics who taught at a nearby university) and suggested we get together to look at what I had come up with.

Naturally he was able to pull three prime numbers out of the air and ask me why the cube values of the two smaller primes, when combined, were larger than the cubed value of the larger prime.

Um … that was embarrassing.

You see, the problem is the so-called “crossover point”.  Are there positive integer values where, when raised to some power greater than 2, their sum equals another positive integer value that is the equivalent power of some other integer?  Does Rn+y + Sn+y = Zn+y?

Technically, there is no difference between this notation and the previous equation I used, except that using them together implies there is some distinctive difference between the two imaginary sets of numbers these equations represent.

Math is filled with proofs that take (a,b,c) and replace them with (x,y,z) or (m,n,q) or both.  It all means something to the mathematicians and the rest of us are left with our jaws hanging out asking, “Um, why couldn’t we just stay with ‘a’, ‘b’, and ‘c’?”

Okay, maybe that is a bit of an oversimplification, but that’s really the point when it comes to Fermat’s Last Theorem.  We don’t know enough math to prove him right using the math he knew.  We had to invent whole new kinds of math just to get to the point where someone, somewhere, could take a wild stab at it and get the whole thing nearly right.

The crossover point is the big problem when it comes to proving Fermat was correct.  If we could show that all equations only had Zsomething that was larger than those other two numbers, someone really smart like Fermat or Euler or one of those math guys from the 18th century would have settled this long ago.  But you get those weird, oddball equations where something doesn’t fit whatever pattern would-be provers-of-Fermat think they have found.

So far you have probably heard all this in some form or another.  Wiles’ proof was accepted by the mathematical community in 1995 but I have continued to play with Fermat in idle moments.  It’s a brain-game for me.  I’m not looking for proofs, I am looking for (new to me) insights into how number systems work.  There are thousands of papers that explain each of the things I discover, usually in some academic gobbledy-gook that many of the academics admit they struggle to understand.

That’s the thing with mathematicians.  They are continually forced to think in mathematical terms and eventually they reach a point where when they try to explain something to anyone else and they sound like idiots and charlatans who escaped from an insane asylum.

Heck, just go read any Wikipedia article that tries to explain advanced math concepts.  They are terribly written because they are not written for people who don’t understand all this stuff.

But what I do understand, myself, is that patterns can usually be identified in all sorts of number systems.  And once in a while I look at some of those patterns and I wonder if Fermat himself was bemused enough by them to think he had found his proof.  Again, I’m not suggesting I think I know what he had in mind.  But he apparently thought it was fairly simple.  Did he have a visual argument that he was never able to articulate?

Here are a couple of charts that illustrate two patterns you can easily see for yourself.  Take a series of squared integers and subtract the lower of two adjacent square numbers from the higher one.  What you’ll get is an odd number and it’s only 2 less than the next difference between two consecutive squares, or 2 higher than the previous one.

Do that same thing with consecutive cubed values and you won’t find any such neat pattern.

But I noticed something interesting: if you draw these series of differences in powers as curves on a chart you will see that the differences in squares have a flat slope whereas the differences in higher powers have a curved slope.

The declining slope of the difference of consecutive squares in a series.

Fig. 1: The flat slope of the difference of consecutive squares in a series vs. the rising slope of the difference of consecutive cubes in a series.

Excel is not exactly the mathematical tool I would prefer to use for making this kind of point but it’s all I have time for as I write this at 1AM in the morning.  Sorry, maths world.  You’ll have to grimace, smirk, and mock me for the inadequacies of my chart.

Another example of an upward slope on the curve of differences in consecutive powers.

Fig. 2: Another example of an upward slope on the curve of differences in consecutive powers.

Is there any significance to the difference in the slopes of these curves?  I dunno.  I’m sure someone somewhere can throw out some gobbledy-gook that mentions Riemann, Taniyama, and Einstein all in 300 pages or less.

I ran the calculation on another couple of series and found that all the higher power curves slope up even more steeply.  The rise in these curves is exponential.  And what this implies is that the crossover point, where the slope of the curve of the differences in consecutive values raised to some power is somewhere in-between the powers of 2 and 3, is a non-integer value.

In other words, there is a fundamental change in the behavior of the slopes of differences of sums of integers raised to the same powers after the differences in squares.

This doesn’t prove that Fermat was right.  It doesn’t show what he was thinking.  But it does illustrate just one of the many amazing qualities of Fermat’s conjecture, which is that there is something different about squared integers that sets them apart from all other integers raised to integer powers.

We are not setting the mathematical world on fire with this revelation tonight but this is the kind of thing that occurs to me all the time because of the work I do.  As a Web marketing analyst I am constantly evaluating data trends.  I look at curves all day long (get your minds out of the gutter).  I spend my days and nights pondering the significance of dips, bumps, wild swings, and sharp jagged points on data lines.

It was my study of curves in data sets that led me to develop the Theory of Deep Web Interferometry, which compares the curves in data trend lines, just as astronomers compare the data in spectral lines.

One cannot help but look for interactions between data sets when dealing with unknowns because it is those interactions that give us clues to what the unknown things are.  In Webometry, especially for marketers, you deal with many unknowns.  The comparative analysis of two or more data trends provides insights into when things happen, although it doesn’t explain what just happened.

In the same way, looking at the slopes of these curves derived from the differences in adjacent integer values computed from integers raised to the same powers reveals that something significant has happened but it doesn’t explain what that significance is.  Our number theory, based on the power of 10, has one oddball set of numbers that is just hard to explain but which has influenced our understanding of mathematics, architecture, art, and many other concepts for thousands of years.

Fig. 3: The classic illustration for the difference in two squares.

Fig. 3: The classic illustration for the difference in two squares.

You can turn up a lot of trivial (insignificant) facts when you play with squared integer values.  Here is another example.  Look at the illustration on the left (not drawn to scale).  This is how we typically illustrate the difference in two squares with geometry.  If you were to draw that illustration on a grid and use values for “a” and “b” that were positive integers, such as 5 and 3, you would be able to rearrange all the units of size (a-b)2 in the dark gray area so that they form a square.

Assuming the white area is always smaller than the dark area, the white area would be 32 (or 9) and the dark area would be 42 (or 16).  But notice the little “a-b” square area in the lower right-hand corner.  Every square integer value can be laid out like this, such that it has three squares associated with it: the inner square (X2), the outer square (Y2), and that little square ((Y-X)2).  No matter how much you adjust the integer value of X, the difference between the integers X and Y will always be a small squared integer value.

You always find this paired relationship (X2 and (Y-X)2) in a difference of squares (using integer values).  You can always find two opposing shapes of different sizes in different corners in higher powers, but there are no sums of two integers that, raised to those powers, produce anything similar to this kind of perfect ratio.

Fermat was definitely on to something, something much larger than he probably realized.  But whether that something will be articulated in a simple, elegant way that everyone can understand remains to be seen.