Tag Archives: prime

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

MaBloWriMo 6: The Proof Begins

Posted on November 7, 2015 by Brent

Today we’re going to start in on proving the Lucas-Lehmer test. Yesterday we saw how, given a Mersenne number , we can define a sequence of integers , with the claim that if and only if is prime. We’re going … Continue reading →

Posted in algebra, arithmetic, computation, number theory, primes | Tagged lehmer, lucas, MaBloWriMo, Mersenne, omega, prime, proof, test | 1 Comment

MaBloWriMo 5: The Lucas-Lehmer Test

Posted on November 6, 2015 by Brent

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

Posted on November 5, 2015 by Brent

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

Posted on November 4, 2015 by Brent

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 →