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Category Archives: primes
A few words about PWW #27
The images in my last post were particular realizations of the famous Sieve of Eratosthenes. The basic idea of the sieve is to repeatedly do the following: Circle the next number bigger than that is not yet crossed out, call … Continue reading
Posted in pattern, pictures, posts without words, primes
Tagged Eratosthenes, prime, sieve
4 Comments
Post without words #27
Posted in pattern, pictures, posts without words, primes
Tagged Eratosthenes, prime, sieve
7 Comments
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
Quickly recognizing primes less than 1000: memorizing exceptional composites
In my previous post I wrote about a procedure for testing the primality of any number less than : Test for divisibility by all primes up to , and also . (In practice I test for 2 and 5 first, … Continue reading
Quickly recognizing primes less than 1000: divisibility tests
I took a little hiatus from writing here since I attended the International Conference on Functional Programming, and since then have been catching up on teaching stuff and writing a bit on my other blog. I gave a talk at … Continue reading
Quickly recognizing primes less than 100
Recently, Mark Dominus wrote about trying to memorize all the prime numbers under . This is a cool idea, but it made me start thinking about alternative: instead of memorizing primes, could we memorize a procedure for determining whether a … Continue reading