Category Archives: fibonacci

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 →

Posted in arithmetic, computation, famous numbers, fibonacci, proof | Tagged , , , , , | 2 Comments

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 →

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

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 →

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

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 →

Posted in arithmetic, computation, famous numbers, fibonacci | Tagged , , , | 8 Comments

Wythoff’s game at Three-Cornered Things

Posted on June 10, 2012 by Brent

I’ve really been enjoying Zachary Abel’s series of posts on Wythoff’s game [Wythoff’s Game: Red or Blue?; A Golden Observation; The “Fibonacci”est String; Wythoff’s Formula], over on his blog Three-Cornered Things. The Fibonacci numbers show up in the strangest places! … Continue reading →

Posted in fibonacci, games, links | Tagged | 7 Comments

Fibonacci multiples, solution 1

Posted on June 9, 2012 by Brent

In a previous post, I challenged you to prove If evenly divides , then evenly divides , where denotes the th Fibonacci number (). Here’s one fairly elementary proof (though it certainly has a few twists!). Pick some arbitrary and … Continue reading →

Posted in fibonacci, modular arithmetic, number theory, pattern, pictures, proof, sequences | Tagged , , , | 5 Comments

Nature by Numbers

Posted on June 6, 2012 by Brent

This has been making the rounds of the math blogosphere (blathosphere?), but in case you haven’t seen it yet, check out Cristóbal Vila’s awesome short video, Nature by Numbers. Especially appropriate given that I have been writing about Fibonacci numbers … Continue reading →

Posted in fibonacci, golden ratio, links, video | Tagged , , , , | 1 Comment

Fibonacci multiples

Posted on May 15, 2012 by Brent

I haven’t written anything here in a while, but hope to write more regularly now that the semester is over—I have a series on combinatorial proofs to finish up, some books to review, and a few other things planned. But … Continue reading →

Posted in arithmetic, challenges, fibonacci, number theory, pattern | Tagged , | 12 Comments

Post without words #1

Posted on May 24, 2011 by Brent
Posted in fibonacci, pattern, pictures, posts without words | Tagged , | 13 Comments

Cassini’s identity

Posted on May 15, 2011 by Brent

My previous post asked you to take any Fibonacci number, square it, and also multiply the two adjacent Fibonacci numbers, and see if a pattern emerged. Here’s a table I made for the first 6 Fibonacci numbers: (Hmm, the numbers … Continue reading →

Posted in algebra, fibonacci, induction, pattern, proof, solutions | Tagged , , | 8 Comments