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# Category Archives: famous numbers

## 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

## 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 Lucas-Lehmer 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 Lucas-Lehmer 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 prime-index 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