Category Archives: iteration

The MacLaurin series for sin(x)

Posted on February 13, 2017 by Brent

In my previous post I said “recall the MacLaurin series for :” Since someone asked in a comment, I thought it was worth mentioning where this comes from. It would typically be covered in a second-semester calculus class, but it’s … Continue reading →

Posted in calculus, infinity, iteration | Tagged , , , , , , , | 4 Comments

MaBloWriMo 16: Recap and outline

Posted on November 17, 2015 by Brent

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 , , , , , , , , , | 2 Comments

MaBloWriMo 8: definition of s and mod

Posted on November 9, 2015 by Brent

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 , , , , , , , | Comments Off on MaBloWriMo 8: definition of s and mod

MaBloWriMo 7: s via omega

Posted on November 8, 2015 by Brent

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 , , , , , , , | 11 Comments

MaBloWriMo 5: The Lucas-Lehmer Test

Posted on November 6, 2015 by Brent

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 →

Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes | Tagged , , , , , , | 3 Comments

MaBloWriMo 4: not all prime-index Mersenne numbers are prime

Posted on November 5, 2015 by Brent

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 , , , , , , | 1 Comment

MaBloWriMo 3: Mersenne composites in binary

Posted on November 4, 2015 by Brent

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 , , , , , , | 1 Comment

MaBloWriMo: Mersenne composites

Posted on November 3, 2015 by Brent

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 , , , , , , | 1 Comment

MaBloWriMo: The Lucas-Lehmer test

Posted on November 2, 2015 by Brent

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 , , , , , , | 5 Comments

Mersenne primes and the Lucas-Lehmer test

Posted on January 17, 2013 by Brent

Mersenne numbers, named after Marin Mersenne, are numbers of the form . The first few Mersenne numbers are therefore , , , , , and so on. Mersenne numbers come up all the time in computer science (for example, is … Continue reading →

Posted in arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes | Tagged , , , , | 3 Comments