# 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

## 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

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## 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

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## 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

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## 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

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## MaBloWriMo 5: The Lucas-Lehmer 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 Lucas-Lehmer test is a way to tell us whether is prime, for any prime . The … Continue reading

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## MaBloWriMo 4: not all prime-index 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

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## 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

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## 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

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## MaBloWriMo: The Lucas-Lehmer 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

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