Fermat’s Last Theorem, the Sum of All Powers
Episode #2 of the course Great math problems for the 21st-century mind by John Robin
Let’s begin today with a simple puzzle.
What do these three equations have in common?
9 + 16 = 25
25 + 144 = 169
49 + 576 = 625
You might have noticed that every number there is a perfect square. That means you can write it as the product of two equal numbers. So, writing this again:
(3)2 + (4)2 = (5)2
(5)2 + (12)2 = (13)2
(7)2 + (24)2 = (25)2
In fact, there are infinitely many examples of these! But what about higher powers than 2? That is to say, what if I tried to add up two cubes? Will I be able to find another number that is a cube, whose sum is made from the other two cubes?
The answer is no. In fact, you can write a cube as the sum of more than two cubes, but never as the sum of only two. You can search and search and search all you want. Add up every cube you can think of—say, 8 + 27—and when you add them up, you will not find a cube.
All right then. Maybe it’s just cubes? What about fourth powers? For example:
(2)4 + (1)4 = (?)4
The answer there is … 17. Not a fourth power. You can try searching all you want, and you wouldn’t be alone in finding no results. In fact, between every mathematician and every computer program in the world, none have ever successfully turned up an example.
And it gets worse. What about fifth powers?
(some number)5 + (some number)5 = (some number)5
Nothing there either. Hmm …
This problem is one that Pierre de Fermat scribbled in the margin of one of his notebooks in the year 1637. His claim was that there are no examples for any higher sum of powers, written in the form:
an + bn = cn
(where the numbers a, b, and c are all natural counting numbers like 0, 1, 2, etc.)
He claimed to have a proof but never wrote it out for us. As he put it, there in the margin where he scribbled it, the proof was “too large to fit in the margin.”
He wasn’t wrong. Fast forward some 350 years, where the field of number theory developed with other fields like group theory, modularity theorem, and elliptic curves, along came that proof. It was 129 pages long—indeed too large to fit in a margin!
Andrew Wiles shared his proof in 1993, but there was a mistake in it. It took him two years to fix it and present the correct one. That’s not to mention several years of work in the individual subfields that had to be developed in order to build the foundation for his proof. In fact, though Andrew Wiles finally is the one to have presented the brilliant conclusion to this long-standing problem, it was the work of numerous mathematicians preceding him, curious about Fermat’s mystery, that led eventually to a proof that could tell us, once and for all, to put our pencils down and stop searching for sums of two cubes that equal a cube and so on.
Tomorrow, we will dig in more fully into this proof and show how one man’s margin note led to a pursuit that would sculpt much of the landscape of modern number theory—indeed, many branches of mathematics that might never have existed if not for the added incentive.
If you want to convince yourself or if you want to dig into the formulas and theories that help you generate examples of equations we talked about in this lesson, this is a great reference.
Number Theory for Beginners by André Weil
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