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.
Now let’s look at what he was actually saying.
Fermat claimed to have proven something fundamental about numbers. He wrote that for any integer* n greater than or equal to 3, there are no positive integers x, y, and z such that
*A positive integer is a whole number bigger than 0. The positive integers are 1, 2, 3, … and so on. Some examples of positive integers are 45 and 8. Some examples of numbers that are not positive integers are ½, -3, 0, and the square root of 7.
This means that, for example, no whole numbers x, y and z will make true x3+y3=z3, or x5+y5=z5, or x21+y21=z21.
You can think of the equation xn+yn=zn as a statement that is always false, as long as x, y, z and n are the types of numbers specified by Fermat’s theorem. An analogous English sentence would be “this watermelon is blue.” No matter which watermelon I am talking about, the sentence is not going to be true.
Quick review: xn can be thought of as “x multiplied by itself n times.” For example, 15=1*1*1*1*1, which is equal to 1. And 32=3*3, which is equal to 9.*
*This is a good way to understand as long as the exponents are positive integers: 1, 2, 3, 4, 5, 6, 7, … and so on. The idea gets a little more complicated if the exponent is a fraction like 1/3, an irrational number like the square root of 2, a negative number, or 0. But Fermat was specifically talking about positive integers greater than or equal to 3. That’s just 3, 4, 5, 6, 7… and so on.
Imagine a slot machine or old-fashioned odometer with slots for x, y, z and n. Spin the x wheel to 3, y wheel to 8, z wheel to 10, and n wheel to 6. That equation is 36+86=106. This is false. It’s easy to check with a calculator. Spin again, and you might get 145+15=95. This is false, too. Try it yourself: pick three positive integers for x, y, and z, and one integer greater than or equal to 3 for n. Plug them all in to Fermat’s theorem.
Is the equation true?
(I’ll bet you a donut it isn’t.)
You may be convinced, after trying enough examples, that there really are no numbers that satisfy the equation.
You could even get a computer to try millions of combinations and check them all. But even with computers, no person has ever found a set of numbers that make true the equation in Fermat’s theorem.
Isn’t that enough? Why did Fermat need proof, if no one can make the equation work?
The need for proof becomes clearer when you consider another possible mathematical claim. In a class with 23 students, there is a fifty-fifty chance that two of them have the same birthday.
Do you think this is true? Take a guess based on your intuition.
Most people think this is false when they first see it. It seems that there should be a much lower chance of two students having the same birthday. Fifty-fifty seems way too likely.
But it is true: the chance that two students have the same birthday is really fifty percent. You can prove it mathematically.
Math is full of weird surprises like this one. Many numerical patterns that seem to be true end up failing. One such pattern (#3 on this list) works for every number up to 8424432925592889329288197322308900672459420460792433.
One simply cannot guess and check such an example.
This brings us back to our question about Fermat. How do we know for sure that there is no example out there that contradicts Fermat’s theorem? What if, when we try 330, 555 and 9834825 to the 90th power, the equation will suddenly turn out true?
There are infinitely many possibilities. We can’t try all of them.
This is where proof comes in. With a proof, we can say, without a doubt, whether Fermat’s Last Theorem is true or false.
But this brings us to an even more basic, fundamental question about math. It is a simple question. A question asked by many. The question is: Why bother?
It’s easy to use something without understanding it. A person can drive a car without knowing how to build an internal combustion engine. A three-year-old can turn on a TV before he even knows the word “electricity.” An engineer can use calculus to design an airplane without knowing how or why calculus is true.
Most of the time, this is fine. But sometimes, a person wants to understand something. It may be something concrete, like why ice melts. Or it may be more abstract, like Van Gogh’s motivation to paint Starry Night. These questions don’t have practical use: it is totally possible to enjoy a cold drink or a painting without deep understanding. But knowledge itself is something people enjoy.
Mathematics is a rich landscape, full of intricate and beautiful ideas. In the last post, we compared a mathematical topic to a cavern of crystals. A proof is like light shed on a crystal. We may have guessed what the crystal would look like, but only once we illuminate it – that is, once we prove a statement once and for all – can we start to appreciate its beauty.
For many people, this is THE reason to do math. It is an incredible experience to “get” something. It is amazing to suddenly see harmony in a complicated structure. People who love math, for the most part, aren’t problem solvers or number crunchers. They are motivated by the desire to know. And math is full of beautiful things to know.