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John Wilson (1741-1793) was a well-known English mathematician in his time, whose legacy lives on in his eponymous result, *Wilson’s Theorem*. To recall, this is the statement that an integer is prime if and only if

(The “if” part is trivial.) As is the case for many historical results, Wilson’s Theorem was *not *proven by Wilson. Instead, it was Joseph Lagrange who provided the first proof.

The proof, as we see it today, might be phrased as follows:

*Proof: *Suppose that is prime. Then each of the nonzero residues modulo is a unit, so that represents the product over all units in . If

, i.e. ,

then and its inverse each show up in our list of units. We cancel out such terms in pairs, and conclude that

We have if and only if , which by primality of forces or . In other words, . It follows that

When is composite, the direct translation of Wilson’s problem gives

The problem, here, is that we’ve multiplied a number of zero divisors together, which can be avoided by only multiplying across the units, , of . In this post, we consider the product

determine its value, give credit to Gauss for doing so over two centuries ago, and discuss a few generalizations.

Back in high school, I came across the following contest problem:

**Question: **Let be a set of positive integers totaling 20. What is the maximum value of

It’s a fun problem, so don’t rush past the spoiler tags too fast. When you’re ready, I’ll spoil the solution to the question above, and discuss a “continuous” version of the question above. Namely, what happens when is allowed to include positive real numbers?

In 1737, Euler presented a novel proof of the infinitude of the primes, in showing that the harmonic sum of the primes diverges. Explicitly, Euler’s work furnishes the estimate

in which the sum exhausts the rational primes at most . At this point, it becomes quite elementary to derive the two inequalities

Results of this flavor remained essentially unimproved for over a century, until Chebyshev presented the following landmark theorem in 1852:

**Theorem (Chebyshev): ***There exist positive constants such that*

Thus Chebyshev’s Theorem shows that represents the growth rate (up to constants) of ; stated equivalently in Bachmann-Landau notation, we have .* *Yet more is true: the constants in Chebyshev’s proof are therein made effective, and can be taken as

As a corollary to Chebyshev’s Theorem, we have for. By making this implicit bound on precise, Chebyshev was able to prove Bertrand’s Postulate (thereafter known as the Bertrand-Chebyshev Theorem).

In this post, we’ll prove a variant of Chebyshev’s Theorem in great generality, and discuss some historically competitive bounds on the constants and given above. Lastly, we’ll discuss how Chebyshev’s Theorem relates to proposed “elementary” proofs of the PNT.

This post serves to resolve some questions posed at the end of a previous article, Unit-Prime Dichotomy for Complex Subrings, in which we began to look at the tension between the relative sizes of the units group and the spectrum of a complex subring (with unity). As a disclaimer, I assume from this point familiarity with the topics presented therein. To begin, we recall a Theorem from the prequel (Theorem 4):

**Theorem: ***Let be a subring (with unity), and suppose that has finite spectrum. Then is dense in , the topological closure of the fraction field of .*

In particular — assuming a finite spectrum — it follows that the rank of (considered as an abelian group) is bounded below by two. Unfortunately, this is as far as our previous methods will take us, even when (see the Exercises).

The primary objective of this post is an extension to this result. Specifically, we would like to capture *in a purely algebraic way* (e.g. without mention of the topology on ) the fact that becomes quite large in certain rings with finite spectrum. After the fold we’ll accomplish exactly that, with the following Theorem and its generalizations to a wide class of commutative rings:

**Theorem 1: ***Suppose that is a subring (with unity). If has finite spectrum, then has infinite rank as an abelian group.*

Throughout, we take as a complex subring (with unity). In this article, we’ll be interested in natural analogues of Euclid’s proof of the infinitude of the primes (i.e. the case . In particular, we’ll show that Euclid’s proof (and generalizations to this method) can be recast as a relationship between the size of the unit group of and , the number of prime ideals in. *(Here, denotes the spectrum of .)*

For the sake of explicit analogy, we include a (needlessly abstracted) version of Euclid’s result now:

**Theorem 1 (Euclid): ** *The integers have infinite spectrum.*

*Proof: *If not, let exhaust the list of prime ideals. As the integers form a PID, (in fact, they form a Euclidean domain), we may associate to each given ideal a prime generator such that . Let ; as is not a unit (replacing with if necessary), it admits a prime factor which equals some . But then divides both and , whence as well. This is a contradiction, and our result follows.

It is worth remarking that the “PID-ness” of the integers is not needed in the derivation of Theorem 1. Indeed, if were not principal, we may take instead to be *any* non-zero element of . Thus, we see that the PID-ness of is *not* the crucial property of the integers (viewed as a subfield of ) upon which our proof stands. Rather — as we’ll see after the fold — Euclid’s proof frames the infinitude of primes as a consequence of the *finiteness of the units group!*

Let be a number field, i.e. a finite field extension to . We recall that the * ring of integers* in

*k*,

*denoted , is the ring*

For , the ring of integers is just the integers , in which case we recall the Fundamental Theorem of Arithmetic: that every integer may be written as a finite product

in which the are prime and uniquely determined (up to permutation). Domains for which this holds are known in general as ** unique factorization domains **(UFDs). For — with square-free — the ring of integers will in general

*not*be a UFD. In fact, for , the integers have unique factorization only in the 9 cases

Far less is known in the case (in which case *k* is known as a * real quadratic field*), although an unproven conjecture dating back to Gauss suggests that there should be infinitely many real quadratic fields. More recently, some heuristics stemming from Cohen suggests that the ring of integers in should be a UFD with probability as on the square-free integers.

Here, we’ll focus on a more tractable variant of this problem:

**Question:** What can be said about the number of distinct real quadratic fields with for which is ** not** a UFD?

For a weak answer to the question above, we devote the rest of this article to the establishment of the following bound:

**Theorem: **As , we have

in which the implied constant is made effective (e.g. greater than ).

In 1832, Galois introduced the concept of normal subgroups, and proved that the groups (for ) and (for ) were ** simple**, i.e. admit no non-trivial (proper) normal subgroups. In 1892, Hölder asked for a classification of all finite simple groups, and the final classification of finite simple groups in 2004 by Aschbacher and Smith’s resolution of the quasithin case thus resolves an open problem over a century in the making.

*(Note: the great majority of the classification theorem concluded in the 1980’s. A computational error was identified and resolved in 2008.)*

One immediate consequence of the classification of finite simple groups (CFSG) is that the set *S* given by

has natural density 0. Yet it seems unlikely that this result requires the *tens of thousands of pages* currently involved in the proof of the CFSG, and for the rest of this article we seek to bound the density of *S* using arithmetic methods.

Let denote the permutation group on *n* letters. Herein, we consider the following question, brought to light by Landau in 1902:

**Question: ***What is the maximal order of an element in ?*

By convention, we shall refer to this maximal order as . Thus, for instance, Lagrange’s theorem gives the elementary bound . We may obtain sharper bounds by noting that is centerless, but tweaks like this will fail to give more than a gain of a multiplicative constant. A far more productive route comes from the following Proposition:

**Proposition:** *Let be a partition of n into positive integers. Then , with equality if and only if for all i.*

*Proof: *Let be a partition of *n* such that is maximized. If for any *i*, we have , and the partition

admits a strictly larger product. Thus, we may assume that for all *i*. Furthermore, the relation implies that in a maximal partition we may have for at most two such *i*. It now follows (by cases) that our maximal product takes the form

Our result follows from the inequalities .

*Note: buried under all of this is the fact that the function attains a global maximum at , and that 3 is the closest integer to e (this also underlies the marginal appearance of 2, as the second closest integer approximation to e). This will become more evident as we continue.*

As a Corollary, we obtain a new upper bound for the function :

**Corollary: ***For all n, we have , with equality if and only if .*

*Proof: *We note that is the maximum value of , as varies over the partitions of *n*. But , with equality if and only if (1) for all *i *and (2) the are pairwise coprime. In particular, equality holds above if and only if (achieved with the partition ).

*Note: this result is stronger than the commonly-cited bound , which holds for all n.*

The bound arising from our Proposition is nevertheless weak in method (in that we have only used the fact that divides ), and to strengthen it significantly requires the Prime number theorem (PNT). In fact, Landau’s classical result

(which we will derive after the fold) is equivalent to the PNT by means of elementary methods.

For what follows, we define a * loaded die *as a discrete probability distribution with six outcomes (labelled {1,…,6}), each of which has positive probability. To each such die, we associate a generating polynomial, given by

in which denotes the probability of the outcome *i*. If corresponds to another such die, we note that the product

has coefficients which reflect the probabilities of certain dice sums for *p* and *q *(and this is the utility of generating polynomials). We are now ready to ask the following question:

**Question: ***Does there exist a pair of loaded dice such that the probability of rolling any dice sum ({2,…12}) is equally likely?*