Category Archives: proof

The Möbius function proof, part 2 (the subset parity lemma)

Continuing from my previous post, we are in the middle of proving that satisfies the same equation as , that is, and that therefore for all , that is, is the sum of all the th primitive roots of unity. … Continue reading

Posted in arithmetic, combinatorics, complex numbers, primes, proof | Tagged , , , , , , , , , | 1 Comment

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 follow-up post, and Post Without Words #11. Recall that we … Continue reading

Posted in geometry, pattern, pictures, posts without words, proof | Tagged , , , , , | 3 Comments

Post without words #11

Posted in geometry, pattern, pictures, posts without words, proof | Tagged , , , , , | 7 Comments

The route puzzle

While poking around some old files I came across this puzzle: (Click for a larger version.) I didn’t make it, and I have no idea where I got it from (do you know?). But in any case, wherever it comes … Continue reading

Posted in arithmetic, challenges, number theory, proof, puzzles | Tagged , , , , , , | 7 Comments

Golden numbers are Fibonacci

This post is fourth in a series, proving the curious fact that is a Fibonacci number if and only if one (or both) of or is a perfect square; we call numbers of this form golden numbers. Last time, I … Continue reading

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Fibonacci numbers are golden

Recall that a “golden number” (this is not standard terminology) is a number such that one (or both) of or is a perfect square. In this post, I’ll explain Gessel’s proof that every Fibonacci number is golden. First, we need … Continue reading

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Testing Fibonacci numbers: the proofs

In my last post I stated this surprising theorem: is a Fibonacci number if and only if one of is a perfect square. If one of is a perfect square, then let’s say that is a “golden number” (a nod, … Continue reading

Posted in arithmetic, computation, famous numbers, fibonacci, proof | Tagged , , , | 1 Comment