Category Archives: algebra

MaBloWriMo 24: Bezout’s identity

A few days ago we made use of Bézout’s Identity, which states that if and have a greatest common divisor , then there exist integers and such that . For completeness, let’s prove it. Consider the set of all linear … Continue reading

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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 20: the group X star

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 19: groups from monoids

So, you have a monoid, that is, a set with an associative binary operation that has an identity element. But not all elements have inverses, so it is not a group. Assuming you really want a group, what can you … Continue reading

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MaBloWriMo 18: X is not a group

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

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 15: One more fact about element orders

I almost forgot, but there is one more fact about the order of elements in a group that we will need. Suppose we have some and we happen to know that is the identity. What can we say about the … Continue reading

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