Tag Archives: omega

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 →

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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 →

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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 →

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MaBloWriMo 17: X marks the spot

Posted on November 18, 2015 by Brent

Recall that we are trying to prove that if is divisible by , then is prime. So let’s suppose is divisible by . We’ll prove this by contradiction, so suppose is not prime: if we can derive a contradiction, then … Continue reading →

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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 , , , , , , , , , | 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 →

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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 , , , , , , , | 1 Comment