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# Category Archives: computation

## Computing the Euler totient function, part 2: seeing phi is multiplicative

In my last post, I claimed that whenever and are relatively prime. (Recall that counts how many numbers from to share no factors in common with .) Let’s get some intuition for this by looking at some Chinese remainder theorem … Continue reading

## Computing the Euler totient function, part 1

Recall that Euler’s totient function counts how many of the integers from to are relatively prime to , that is, share no factors in common with . For example, , since only , , , and share no factors with … Continue reading

## Goldilogs and the n bears

Once upon a time there was a girl named Goldilogs. As she was walking through the woods one day, she came upon a curious, long house. Walking all round it and seeing no one at home, she tried the door … Continue reading

## Finding the repetend length of a decimal expansion

We’re still 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 repeating part. In my previous … Continue reading

Posted in computation, group theory, modular arithmetic, number theory, pattern
Tagged decimal, expansion, group theory, rational, repeating, repetend, totient

## 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
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