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Tag Archives: theorem
Chinese Remainder Theorem proof
In my previous post I stated the Chinese Remainder Theorem, which says that if and are relatively prime, then the function is a bijection between the set and the set of pairs (remember that the notation means the set ). … Continue reading
More words about PWW #25: The Chinese Remainder Theorem
In a previous post I made images like this: And then in the next post I explained how I made the images: starting in the upper left corner of a grid, put consecutive numbers along a diagonal line, wrapping around … Continue reading
Posted in modular arithmetic, number theory, posts without words
Tagged Chinese, grid, remainder, theorem, torus
3 Comments
A few words about PWW #25
In my previous post I made images like this: What’s going on? Well, first, it’s easy to notice that each grid starts with in the upperleft square; is one square down and to the right of , then is one … Continue reading
Posted in modular arithmetic, number theory, posts without words
Tagged Chinese, grid, remainder, theorem, torus
4 Comments
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 combinatorics, counting, Fermat, little, necklace, proof, theorem
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
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
Four formats for Fermat: correction!
In my previous post I explained three variants on Fermat’s Little Theorem, as well as a fourth, slightly more general variant, which it turns out is often called Euler’s Totient Theorem. Here’s what I said: If and is any integer, … Continue reading
Posted in number theory, primes
Tagged correction, Euler, Fermat, little, prime, theorem, totient
4 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 publickey cryptography and primality testing. (It’s called “little” to distinguish it from his (in)famous Last Theorem.) There are … Continue reading
Apollonian gaskets and Descartes’ Theorem II
In a few previous posts I wrote about “kissing sets” of four mutually tangent circles, and the fact that their signed bends satisfy Descartes’ Theorem, (Remember that the signed bend of a circle is like the curvature , except that … Continue reading