### Meta

### Categories

- algebra (46)
- arithmetic (70)
- books (30)
- calculus (7)
- challenges (54)
- combinatorics (21)
- complex numbers (6)
- computation (67)
- convergence (9)
- counting (32)
- famous numbers (48)
- fibonacci (18)
- fractals (13)
- games (34)
- geometry (71)
- golden ratio (8)
- group theory (27)
- humor (6)
- induction (7)
- infinity (19)
- iteration (24)
- links (76)
- logic (7)
- meta (43)
- modular arithmetic (24)
- number theory (91)
- open problems (11)
- paradox (1)
- pascal's triangle (8)
- pattern (94)
- people (21)
- pictures (70)
- posts without words (27)
- primes (48)
- probability (6)
- programming (18)
- proof (83)
- puzzles (16)
- recursion (16)
- review (21)
- sequences (28)
- solutions (28)
- teaching (14)
- trig (3)
- Uncategorized (6)
- video (19)

### Archives

- September 2018 (4)
- August 2018 (6)
- July 2018 (2)
- June 2018 (5)
- May 2018 (3)
- April 2018 (5)
- March 2018 (4)
- February 2018 (3)
- January 2018 (4)
- December 2017 (3)
- November 2017 (3)
- October 2017 (1)
- September 2017 (1)
- July 2017 (4)
- June 2017 (4)
- May 2017 (9)
- April 2017 (7)
- March 2017 (5)
- February 2017 (4)
- January 2017 (3)
- December 2016 (4)
- November 2016 (6)
- October 2016 (6)
- September 2016 (2)
- August 2016 (5)
- July 2016 (2)
- June 2016 (4)
- May 2016 (4)
- April 2016 (2)
- March 2016 (3)
- February 2016 (9)
- January 2016 (8)
- December 2015 (5)
- November 2015 (29)
- August 2015 (3)
- June 2015 (2)
- April 2015 (1)
- May 2014 (1)
- December 2013 (1)
- October 2013 (1)
- July 2013 (1)
- June 2013 (1)
- May 2013 (1)
- April 2013 (3)
- March 2013 (3)
- February 2013 (2)
- January 2013 (5)
- December 2012 (3)
- November 2012 (4)
- October 2012 (5)
- September 2012 (1)
- August 2012 (4)
- July 2012 (1)
- June 2012 (6)
- May 2012 (2)
- April 2012 (3)
- March 2012 (1)
- February 2012 (4)
- January 2012 (5)
- December 2011 (1)
- November 2011 (7)
- October 2011 (4)
- September 2011 (6)
- July 2011 (2)
- June 2011 (4)
- May 2011 (5)
- April 2011 (2)
- March 2011 (4)
- February 2011 (1)
- January 2011 (1)
- December 2010 (1)
- November 2010 (4)
- October 2010 (2)
- September 2010 (1)
- August 2010 (1)
- July 2010 (1)
- June 2010 (2)
- May 2010 (3)
- April 2010 (1)
- February 2010 (6)
- January 2010 (3)
- December 2009 (8)
- November 2009 (7)
- October 2009 (3)
- September 2009 (3)
- August 2009 (1)
- June 2009 (4)
- May 2009 (5)
- April 2009 (4)
- March 2009 (2)
- February 2009 (1)
- January 2009 (7)
- December 2008 (1)
- October 2008 (2)
- September 2008 (7)
- August 2008 (1)
- July 2008 (1)
- June 2008 (1)
- April 2008 (5)
- February 2008 (4)
- January 2008 (4)
- December 2007 (3)
- November 2007 (12)
- October 2007 (2)
- September 2007 (4)
- August 2007 (3)
- July 2007 (1)
- June 2007 (3)
- May 2007 (1)
- April 2007 (4)
- March 2007 (3)
- February 2007 (7)
- January 2007 (1)
- December 2006 (2)
- October 2006 (2)
- September 2006 (6)
- July 2006 (4)
- June 2006 (2)
- May 2006 (6)
- April 2006 (3)
- March 2006 (6)

# 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, 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 fibonacci, formula, square, test
8 Comments

## Wythoff’s game at Three-Cornered 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 Three-Cornered 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