Category Archives: arithmetic

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

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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

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The wizard’s rational puzzle (solutions, part 2)

At long last, here is the solution I had in mind for the Wizard’s rational puzzle. Recall that the goal is to figure out the numerator and denominator of a secret rational number, if all we are allowed to do … Continue reading

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The wizard’s rational puzzle (solutions, part 1)

About two and a half months ago I posted a challenge involving a sadistic math wizard, metal cubes containing rational numbers, and a room full of strange machines. I’ve been remiss in following up with some solutions. (Go read the … Continue reading

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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

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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

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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

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