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Tag Archives: Fermat
More on Fermat witnesses and liars
In my previous post I stated, without proof, the following theorem: Theorem: if is composite and there exists at least one Fermat witness for , then at least half of the numbers relatively prime to are Fermat witnesses. Were you … Continue reading
Posted in computation, number theory, primes
Tagged Carmichael, Fermat, liar, primality, test, witness
Comments Off on More on Fermat witnesses and liars
Fermat witnesses and liars (some words on PWW #24)
Let be a positive integer we want to test for primality, and suppose is some other positive integer with . There are then four possibilities: and could share a common factor. In this case we can find the common factor … Continue reading
Posted in computation, number theory, posts without words, primes
Tagged Fermat, liar, primality, test, witness
1 Comment
Post without words #24
Posted in computation, number theory, posts without words, primes
Tagged Carmichael, Fermat, primality, test
5 Comments
The Fermat primality test and the GCD test
In my previous post we proved that if shares a nontrivial common factor with , then , and this in turn proves that is not prime (by Fermat’s Little Theorem). But wait a minute, this is silly: if shares a … Continue reading
Making the Fermat primality test deterministic
Let’s recall Fermat’s Little Theorem: If is prime and is an integer where , then . Recall that we can turn this directly into a test for primality, called the Fermat primality test, as follows: given some number that we … Continue reading
Posted in computation, number theory, primes
Tagged deterministic, Fermat, primality, test
1 Comment
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
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