Tag Archives: X

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 →

Posted in algebra, group theory, modular arithmetic, number theory, proof | Tagged , , , , , , , , , , | 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 →

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MaBloWriMo 20: the group X star

Posted on November 21, 2015 by Brent

So, where are we? Recall that we are assuming (in order to get a contradiction) that is not prime, and we picked a smallish divisor (“smallish” meaning ). We then defined the set as that is, combinations of and where … Continue reading →

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MaBloWriMo 18: X is not a group

Posted on November 19, 2015 by Brent

Yesterday we defined along with a binary operation which works by multiplying and reducing coefficients . So, is this a group? Well, let’s check: It’s a bit tedious to prove formally, but the binary operation is in fact associative. Intuitively … 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 →

Posted in algebra, arithmetic, group theory, number theory | Tagged , , , , | 3 Comments