Tag Archives: lucas

Golden numbers are Fibonacci

Posted on March 19, 2016 by Brent

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

Posted on February 28, 2016 by Brent

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

M74207281 is prime!

Posted on January 24, 2016 by Brent

I’m a bit late to the party, but the Great Internet Mersenne Prime Search has recently announced a newly verified prime number, , with a whopping 22,338,618 decimal digits! This is now the largest known prime number (though of course … Continue reading →

Posted in computation | Tagged lehmer, lucas, Mersenne, new, prime | Comments Off

MaBloWriMo 23: contradiction!

Posted on November 24, 2015 by Brent

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

Posted on November 23, 2015 by Brent

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

Posted on November 22, 2015 by Brent

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

MaBloWriMo 16: Recap and outline

Posted on November 17, 2015 by Brent

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 9: omega and its ilk

Posted on November 10, 2015 by Brent

So far, we have defined a sequence of numbers , and showed that where and . This is a big step: the are defined recursively (that is, each is defined in terms of the previous ), but and give us … Continue reading →

Posted in algebra, arithmetic, modular arithmetic, number theory | Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, prime | Comments Off

MaBloWriMo 8: definition of s and mod

Posted on November 9, 2015 by Brent

I was a little unsatisfied with my proof yesterday since I don’t think I did a very good job explaining how enters into things. When sinuheancelmo asked a question which seemed to show confusion on exactly that point, I figured … Continue reading →

Posted in arithmetic, iteration, modular arithmetic, number theory | Tagged definition, lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test | Comments Off

MaBloWriMo 7: s via omega

Posted on November 8, 2015 by Brent

Yesterday, I challenged you to prove that where , , and the are defined by and . The proof is by induction on . The base case is just arithmetic: Now suppose that we already know the statement holds for … Continue reading →

Posted in algebra, arithmetic, iteration, modular arithmetic, number theory | Tagged induction, lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test | 11 Comments