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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

## 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

## 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

## 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