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Category Archives: famous numbers
Happy tau day!
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
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
MaBloWriMo 16: Recap and outline
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
Posted in algebra, arithmetic, computation, famous numbers, group theory, iteration, modular arithmetic, number theory, primes
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, prime, proof, summary, test
2 Comments
MaBloWriMo 5: The LucasLehmer Test
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 LucasLehmer test is a way to tell us whether is prime, for any prime . The … Continue reading
Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
3 Comments
MaBloWriMo 4: not all primeindex Mersenne numbers are prime
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
Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
1 Comment
MaBloWriMo 3: Mersenne composites in binary
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
Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
1 Comment
MaBloWriMo: Mersenne composites
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 lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
1 Comment