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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
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 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
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
Posted in algebra, arithmetic, computation, famous numbers, group theory, iteration, modular arithmetic, number theory, primes
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, prime, proof, summary, test
2 Comments
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
Posted in algebra, arithmetic, modular arithmetic, number theory
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, prime
Comments Off on MaBloWriMo 9: omega and its ilk
MaBloWriMo 6: The Proof Begins
Today we’re going to start in on proving the LucasLehmer 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
Posted in algebra, arithmetic, computation, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, omega, prime, proof, test
1 Comment