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# Tag Archives: test

## 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 23: contradiction!

So, where are we? We assumed that is divisible by , but is not prime. We picked a divisor of and used it to define a group , and yesterday we showed that has order in . Today we’ll use … Continue reading

Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged contradiction, groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
5 Comments

## MaBloWriMo 22: the order of omega, part II

Yesterday, from the assumption that is divisible by , we deduced the equations and which hold in the group . So what do these tell us about the order of ? Well, first of all, the second equation tells us … Continue reading

Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
1 Comment

## MaBloWriMo 21: the order of omega, part I

Now we’re going to figure out the order of in the group . Remember that we started by assuming that passed the Lucas-Lehmer test, that is, that is divisible by . Remember that we also showed for all . In … Continue reading

Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
2 Comments