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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 wellknown 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