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# Category Archives: number theory

## Finding the prefix length of a decimal expansion

Remember from my previous post that we’re trying to find the prefix length and repetend length of the decimal expansion of a fraction , that is, the length of the part before it starts repeating, and the length of the … Continue reading

## Finding prefix and repetend length

We interrupt your regularly scheduled primality testing to bring you something else fun I’ve been thinking about. It’s well-known that any rational number has a decimal expansion that either terminates, or is eventually periodic—that is, the digits after the decimal … Continue reading

## 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

## Primality testing: recap

Whew, this is developing into one of the longest post series I’ve ever written (with quite a few tangents and detours along the way). I thought it would be worth taking a step back for a minute to recap what … Continue reading

## Post without words #22

Posted in arithmetic, computation, number theory, posts without words
Tagged logarithmic, repeated, squaring
8 Comments

## Modular exponentiation by repeated squaring

In my last post we saw how to quickly compute powers of the form by repeatedly squaring: ; then ; and so on. This is much more efficient than computing powers by repeated multiplication: for example, we need only three … Continue reading

Posted in computation, number theory
Tagged algorithm, exponentiation, logarithmic, modular, primality, repeated, squaring, test
4 Comments