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 . 308 more words

## Tags » Mersenne

#### MaBloWriMo 24: Bezout’s identity

#### 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… 320 more words

#### 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 ? 196 more words

#### 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 . 300 more words

#### 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 ). 134 more words

#### 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. 288 more words

#### 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. 185 more words