Odd perfect numbers: lower bound

The first few even perfect numbersA perfect number is a positive integer $n$ such that $n$ is the sum of its proper divisors. For example $6 = 1 + 2 + 3$. The symbol $\sigma(n)$ is usally used for the sum of all the divisors of a positive integer $n$, so that a number is perfect if and only if $\sigma(n) = 2n$. All known perfect numbers are even, and they correspond to Mersenne primes. These are primes of the form $2^k – 1$. For example, if $k=5$ then $2^5 – 1 = 31$, a prime number. The correspondence between Mersenne primes and even perfect numbers is given by:
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Graph: number of primes containing a given digit

You can ask lots of questions about primes. After posting 50 facts about primes, I couldn't resist making another graph. In this one, the x-axis is $n$ and the y-axis is the number of primes up to $n$ that contain a given decimal digit (written in decimal, of course). I've plotted all of these on a single graph, with different colours for each digit. It contains the first thousand primes.
Keep in mind that these categories of primes are not mutually exclusive. There are some trends here that are not surprising: the after the prime 5, prime numbers can only end in 1, 3, 7, or 9, since the other endings have a common factor with 10. It would probably be difficult to prove that any one function dominates another.

50 Awesome facts about prime numbers

A prime is a natural number greater than one whose only factors are one and itself. I find primes pretty cool, so I made a list of 50 facts about primes:

  1. The first twenty primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.
  2. For any positive natural number $n$, there is always a prime between $n$ and $2n$. This is known as Bertrand's postulate.
  3. A Mersenne prime is a prime of the form $2^n – 1$. Such a number is prime if and only if $2^{n-1}(2^n-1)$ is a perfect number (a number equal to the sum of all its proper divisors). All even perfect numbers arise this way, and no one knows if an odd perfect number exists.
  4. Mersenne would be pretty thrilled that people keep looking for his primes

  5. There are 51 known Mersenne primes, and it is unknown whether there are infinitely many.
  6. The largest prime that humans have ever found is $2^{82,589,933}-1$ and is a Mersenne prime. It was discovered in 2018.
  7. It is pretty easy to come up with a new kind of prime, such as a "palindromic prime", but most of the time it is very difficult to prove that there are infinitely many such primes. In case you're curious, 142840628019121910826048241 is a palindromic prime.
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Exotic dimensions used in ring theory

Do you ever get the feeling that mathematics uses the word dimension a lot? Well, that's for good reason. The concept of dimension is fundamental in mathematics. What is dimension? You can think of dimension as a numerical invariant characterizing the number of parameters required to do a certain thing. For example, for vector spaces, dimension is the cardinality of a basis, and a basis is a minimal set from which you can specify all vectors via linear combinations. The cartesian plane is two-dimensional because you need two coordinates in order to specify any point.

There are other more exotic types of dimension used in ring theory, and this post aims to be a quick introduction to them.

Rank of a free module

Possible values: all cardinals.

As we've already talked about, the dimension of a vector space is the cardinality of a basis of that space. Every vector space has a basis, and all bases of a given vector space have the same cardinality. Therefore, the concept of dimension in this case is well-defined.
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Beamer vs reveal.js for math presentations

I've used Beamer to prepare all my slide-based math presentations, and so does virtually everyone else. It works pretty well with minimal effort. It even has sensible defaults to dissuade users from creating walls of text, although I've definitely seen my share of walls of text.

Recently there has been a craze of JavaScript-powered presentation frameworks, and I decided to try reveal.js. To get them to work, you have to drop a bunch of files in a directory, edit the presentation HTML file, and open it with a browser. The browser will then run some Javascript and display the presentation. The easiest way to see a demo is just visit the link I just gave.

In my trial I created a few presentations with Reveal to see whether it could replace Beamer. Some of you are probably asking why I would even want to do that, given that Beamer works so well. Actually, I was just curious. However, I also found that Beamer is difficult to tweak when the need arises. Modifying themes and customizing the layout of slides is not easy. That's not Beamer's fault. Pretty much all of LaTeX follows a simple pattern: if something doesn't work, look on StackExchange. Seriously: TeX is a baroque language. On the other hand, I somehow doubt anything could ever replace it. The annoyances that do occur are minor, it is practically bug-free, and it is so stable that I'm sure the source files I've already created will still compile into identical PDFs long after the heat death of the universe.
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Free notes on rigidity of groups acting on manifolds

A final version of a 160-page text written by Aaron Brown and others appeared on the arXiv today. The abstract:

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teresópolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory — particularly Lyapunov exponents and metric entropy — as tools to study rigidity properties of group actions on manifolds.
We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems 16, 1996] and recent the work of the main author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [arXiv:1608.04995; arXiv:1610.09997]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures.

Weak dimension one rings are axiomatizable

Let $R$ be a ring. In the previous post on pure exact sequences, we called an exact sequence $0\to A\to B\to C\to 0$ of left $R$-modules pure if its image under any functor $X\otimes -$ is an exact sequence of abelian groups for any right $R$-module $X$. Here is yet another characterization of purity:

Theorem. Let $R$ be a ring. An exact sequence of $R$-modules $0\to A\to B\to C\to 0$ is pure if and only if for every diagram
there exists an $R$-module homomorphism $\theta:R^m\to A$ such that $\theta\sigma=\alpha$.

I won't prove this here since an excellent exposition can be found in
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Pure exact sequences

Over the next few posts, I'll talk more about axiomatizability of algebraic structures in first-order logic. Before I do that, we need to know about purity of exact sequences. So let's fix a ring $R$. An exact sequence
$$0\to A\to B\to C\to 0$$ in the category of left $R$ modules is called pure if for every right $R$-module $C'$, the sequence
$$0\to A\otimes_R C'\to B\otimes_R C'\to C\otimes_R C'\to 0$$ is exact. Notice how this is like a dual concept to flatness: a right $R$-module is flat if its associated tensor functor preserves every exact sequence in the category of left $R$-modules. Whereas, a sequence is pure if its preserved by every tensor product functor.

However, it turns out we can also characterize flatness in terms of purity.

Theorem. A left $R$-module $C$ is flat if and only if every exact sequence of the form $0\to A\to B\to C\to 0$ is pure.

This is quite interesting. In other words, this theorem says that flat modules are precisely those modules that appear as quotients exclusively in pure exact sequences. Let's see why this is true.
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Axiomatizability of classes of structures

Let's talk about axiomatizability in first-order logic, with examples in ring theory. Let's call a class $\Ccl$ of rings axiomatizable if there exists a set $T$ of first order sentences such that $C\in\Ccl$ if and only if $C$ is a model of $T$ (that is, satisfies every sentence in $T$.)

What are some examples? The class of all rings is axiomatizable, because a rings are defined by a set of first-order axioms! The class of commutative rings is also axiomatizable because it is the class of rings that satisfies the additional sentence
$$\forall x\forall y(xy = yx).$$The class of fields is also axiomatizable, since it is the class of commutative rings that satisfies the additional sentence
$$\forall x(\lnot(x=0)\rightarrow \exists y(xy = 1)).$$ In fact, many classes of rings are axiomatizable. Try and think of a few more. On the other hand, there are some examples that are just not axiomatizable. For example, if you take any infinite ring $R$, then the class $\Ccl$ of all rings isomorphic to $R$ is not axiomatizable. Why is this? It is because if you have an infinite model of a set of sentences, then there exists models of arbitrary infinite cardinality, but every ring in $\Ccl$ has the same cardinality as $R$.

When is a class of rings is axiomatizable? Here is one set of criteria:
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Fun with principal ideal domains

A commutative ring $R$ is called a principal ideal domain (PID) if every ideal of $R$ can be generated by a single element. If $R$ is a principal ideal domain, is every subring of $R$ a principal ideal domain? No, definitely not. That is because you can take any integral domain that is not a principal ideal domain, like $\Z[x]$, and take its fraction field. Its fraction field is a PID and the original ring sits inside it as a subring.

Another more interesting example is the ring $\Q[x]$ of polynomials with rational coefficients. It is a PID, yet the subring $\Q[x^2,x^3,x^4,\dots]$ is not. The ideal $(x^2,x^3)$ in this ring is not a principal ideal. By the way, is the ring $\Q[x^2,x^3,\dots]$ Noetherian? Does there exist an ideal in it that needs at least three generators?
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