Category Archives: famous numbers

Happy tau day!

Posted on June 28, 2021 by Brent

Happy Tau Day! The Tau Manifesto by Michael Hartl is now available in print, if you’re in the market for a particularly nerdy coffee table book and conversation starter: I also wrote about Tau Day ten years ago; see that … Continue reading →

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Golden numbers are Fibonacci

Posted on March 19, 2016 by Brent

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

Posted on February 28, 2016 by Brent

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

Posted on February 27, 2016 by Brent

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 →

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Testing Fibonacci numbers

Posted on February 26, 2016 by Brent

From a recent post on Brian Hayes’ blog, bit-player, I learned the following curious fact: is a Fibonacci number if and only if either or is a perfect square. Recall that the Fibonacci numbers begin where each number is the … Continue reading →

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MaBloWriMo 16: Recap and outline

Posted on November 17, 2015 by Brent

We have now established all the facts we will need about groups, and have incidentally just passed the halfway point of MaBloWriMo. This feels like a good time to take a step back and outline what we’ve done so far … Continue reading →

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MaBloWriMo 5: The Lucas-Lehmer Test

Posted on November 6, 2015 by Brent

We now know that can only be prime when is prime; but even when is prime, sometimes is prime and sometimes it isn’t. The Lucas-Lehmer test is a way to tell us whether is prime, for any prime . The … Continue reading →

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MaBloWriMo 4: not all prime-index Mersenne numbers are prime

Posted on November 5, 2015 by Brent

Over the past couple days we saw that if is composite, then is also composite. Equivalently, this means that if we want to be prime, then at the very least must also be prime. But at this point there is … Continue reading →

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MaBloWriMo 3: Mersenne composites in binary

Posted on November 4, 2015 by Brent

Yesterday we saw that must be composite, since . Today I’ll talk about a somewhat more intuitive way to see this. Recall that we can write numbers in base 2, or “binary”, using the digits 0 and 1 (called “bits”, … Continue reading →

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MaBloWriMo: Mersenne composites

Posted on November 3, 2015 by Brent

The name of the game is to find Mersenne numbers which are also prime. Today, a simple observation: can only be prime when is also prime. Put conversely, if is composite then is also composite. For example, is composite and … Continue reading →

Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes | Tagged , , , , , , | 1 Comment