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Category Archives: modular arithmetic
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 11: Examples of Groups
For reference, here’s the definition of a group again: a set a special element a binary operation on such that is associative, that is, whenever , , and are elements of is the identity for , that is, for every … Continue reading
Posted in algebra, group theory, modular arithmetic, number theory
Tagged examples, groups, MaBloWriMo
5 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 8: definition of s and mod
I was a little unsatisfied with my proof yesterday since I don’t think I did a very good job explaining how enters into things. When sinuheancelmo asked a question which seemed to show confusion on exactly that point, I figured … Continue reading
Posted in arithmetic, iteration, modular arithmetic, number theory
Tagged definition, lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
Comments Off on MaBloWriMo 8: definition of s and mod
MaBloWriMo 7: s via omega
Yesterday, I challenged you to prove that where , , and the are defined by and . The proof is by induction on . The base case is just arithmetic: Now suppose that we already know the statement holds for … Continue reading
Posted in algebra, arithmetic, iteration, modular arithmetic, number theory
Tagged induction, lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
11 Comments
MaBloWriMo 5: The LucasLehmer Test
We now know that can only be prime when is prime; but even when is prime, sometimes is prime and sometimes it isn’t. The LucasLehmer test is a way to tell us whether is prime, for any prime . The … Continue reading
Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
3 Comments
MaBloWriMo 4: not all primeindex Mersenne numbers are prime
Over the past couple days we saw that if is composite, then is also composite. Equivalently, this means that if we want to be prime, then at the very least must also be prime. But at this point there is … Continue reading
Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
1 Comment
MaBloWriMo 3: Mersenne composites in binary
Yesterday we saw that must be composite, since . Today I’ll talk about a somewhat more intuitive way to see this. Recall that we can write numbers in base 2, or “binary”, using the digits 0 and 1 (called “bits”, … Continue reading
Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
1 Comment
MaBloWriMo: Mersenne composites
The name of the game is to find Mersenne numbers which are also prime. Today, a simple observation: can only be prime when is also prime. Put conversely, if is composite then is also composite. For example, is composite and … Continue reading
Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
1 Comment
MaBloWriMo: The LucasLehmer test
Today, I noticed both Zachary Abel and Qiaochu Yuan plan to write a blog post every day this month (hooray!). I haven’t written on here as much as I would like recently, and so I thought, why not? I already … Continue reading
Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
5 Comments