Category Archives: proof

The chromatic number of the plane, part 1

About a week ago, Aubrey de Grey published a paper titled “The chromatic number of the plane is at least 5”, which is a really cool result. It’s been widely reported already, so I’m actually a bit late to the … Continue reading

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Properties of orthogons I

First things first: from now on, when talking about polygons with only right angles, instead of calling them “orthogonal polygons” I’m going to start calling them “orthogons”, which sounds cool, is much less clunky than “orthogonal polygons”, and doesn’t seem … Continue reading

Posted in combinatorics, geometry, proof | Tagged , , , , , , , | 10 Comments

Fermat’s Little Theorem: proof by group theory

It’s time for our third and final proof of Fermat’s Little Theorem, this time using some group theory. This proof is probably the shortest—explaining this proof to a professional mathematician would probably take only a single sentence—but requires you to … Continue reading

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Fermat’s Little Theorem: proof by necklaces

It’s time for our second proof of Fermat’s Little Theorem, this time using a proof by necklaces. As you know, proof by necklaces is a very standard technique for… wait, what do you mean, you’ve never heard of proof by … Continue reading

Posted in combinatorics, number theory, primes, proof | Tagged , , , , , , | 4 Comments

Euler’s Theorem: proof by modular arithmetic

In my last post I explained the first proof of Fermat’s Little Theorem: in short, and hence . Today I want to show how to generalize this to prove Euler’s Totient Theorem, which is itself a generalization of Fermat’s Little … Continue reading

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Fermat’s Little Theorem: proof by modular arithmetic

In a previous post I explained four (mostly) equivalent statements of Fermat’s Little Theorem (which I will abbreviate “FlT”—not “FLT” since that usually refers to Fermat’s Last Theorem, whose proof I am definitely not qualified to write about!). Today I … Continue reading

Posted in number theory, primes, proof | Tagged , , | 8 Comments

Four formats for Fermat

In my previous post I mentioned Fermat’s Little Theorem, a beautiful, fundamental result in number theory that underlies lots of things like public-key cryptography and primality testing. (It’s called “little” to distinguish it from his (in)famous Last Theorem.) There are … Continue reading

Posted in number theory, primes, proof | Tagged , , | 3 Comments