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Category Archives: number theory
Modular exponentiation
In my previous post I explained the Fermat primality test: Input: Repeat times: Randomly choose . If , stop and output COMPOSITE. Output PROBABLY PRIME. In future posts I’ll discuss how well this works, things to worry about, and so … Continue reading
The Fermat primality test
After several long tangents writing about orthogons and the chromatic number of the plane, I’m finally getting back to writing about primality testing. All along in this series, my ultimate goal has been to present some general primality testing algorithms … Continue reading
Fast and slow machines
In my previous post, I presented three hypothetical machines which take a positive integer as input and give us something else as output: a factorization machine gives us the complete prime factorization of ; a factor machine gives us one … Continue reading
Posted in computation, number theory, primes
Tagged algorithm, division, efficiency, exponential, factor, factorization, machine, polynomial, primality, testing, trial
1 Comment
New Mersenne prime
With impeccable timing, just in the middle of my series about primality testing, a new Mersenne prime has been announced, a little under two years after the previous one. In particular, it has been shown that is prime; this is … Continue reading
A tale of three machines
The Fundamental Theorem of Arithmetic tells us that any positive integer can be factored into a product of prime factors.1 Given a positive integer , this leads naturally to several questions: What is the prime factorization of ? This is … Continue reading
Posted in computation, number theory, primes
Tagged factor, factorization, machine, primality, testing
11 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
Posted in group theory, number theory, primes, proof
Tagged combinatorics, group, order, proof, theory
Comments Off on Fermat’s Little Theorem: proof by group theory
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