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Tag Archives: X
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
Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged contradiction, groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
5 Comments
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
Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
1 Comment
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 LucasLehmer test, that is, that is divisible by . Remember that we also showed for all . In … Continue reading
Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
2 Comments
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
Posted in algebra, arithmetic, group theory, number theory
Tagged groups, MaBloWriMo, monoids, X
Comments Off on MaBloWriMo 20: the group X star
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
Posted in algebra, arithmetic, group theory, number theory
Tagged groups, MaBloWriMo, monoids, X
1 Comment
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
Posted in algebra, arithmetic, group theory, number theory
Tagged groups, MaBloWriMo, omega, proof, X
3 Comments