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Tag Archives: Euler
Computing the Euler totient function, part 4: totient of prime powers
I’ve been on a bit of a hiatus as I’ve been travelling with my family for the past month. So here’s a recap. Our story so far Recall that the Euler totient function, , counts how many numbers from to … Continue reading
Posted in arithmetic, computation, counting
Tagged Euler, multiplicative, totient
Comments Off on Computing the Euler totient function, part 4: totient of prime powers
Computing the Euler totient function, part 3: proving phi is multiplicative
We are trying to show that the Euler totient function , which counts how many numbers from to share no common factors with , is multiplicative, that is, whenever and share no common factors. In my previous post, we looked … Continue reading
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
Post without words #26
Posted in modular arithmetic, number theory, posts without words
Tagged Euler, grid, totient
2 Comments
Euler’s Theorem: proof by modular arithmetic
In my last post I explained the first proof of Fermat’s Little Theorem: in short, and hence . Today I want to show how to generalize this to prove Euler’s Totient Theorem, which is itself a generalization of Fermat’s Little … Continue reading
Four formats for Fermat: correction!
In my previous post I explained three variants on Fermat’s Little Theorem, as well as a fourth, slightly more general variant, which it turns out is often called Euler’s Totient Theorem. Here’s what I said: If and is any integer, … Continue reading
Posted in number theory, primes
Tagged correction, Euler, Fermat, little, prime, theorem, totient
4 Comments
Totient sums
I took a bit of a break to travel to Japan for a conference, but I’m back now to continue the series I started with Post Without Words #10, a followup post, and Post Without Words #11. Recall that we … Continue reading
Hyperbinary conjecture seeking proof for a good time, long walks on the beach
Here’s the latest progress on the hyperbinary sequence. We’re trying to figure out the inverse relation of the function : given a particular number , where does it occur in the hyperbinary sequence? That is, what are the values of … Continue reading
Posted in challenges, pattern, people, proof, sequences
Tagged Euler, hyperbinary, inverse relation, totient
4 Comments