Tag Archives: omega

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

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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

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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

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MaBloWriMo 17: X marks the spot

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

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

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MaBloWriMo 9: omega and its ilk

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

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

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