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Category Archives: fibonacci
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
Posted in arithmetic, computation, famous numbers, fibonacci, proof
Tagged Cassini, fibonacci, formula, lucas, square, test
2 Comments
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
Posted in arithmetic, computation, famous numbers, fibonacci, proof
Tagged Cassini, fibonacci, formula, lucas, square, test
1 Comment
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 fibonacci, formula, square, test
1 Comment
Testing Fibonacci numbers
From a recent post on Brian Hayes’ blog, bitplayer, 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 fibonacci, formula, square, test
8 Comments
Wythoff’s game at ThreeCornered Things
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 ThreeCornered Things. The Fibonacci numbers show up in the strangest places! … Continue reading
Fibonacci multiples, solution 1
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 divisibility, fibonacci, proof, remainders
5 Comments
Nature by Numbers
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
Fibonacci multiples
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 divisibility, fibonacci
12 Comments
Cassini’s identity
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