Short papers for undergraduate course on reading scholarly math

Question

(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.)

Today, I was reminded of the existence of this paper: Terminating Decimals in the Cantor Set.

It is a concise paper (3.5 pages) that employs nothing too sophisticated, just some modular arithmetic and careful casing. I thought it would be a wonderful to spend a few class meetings with undergraduate math majors reading this paper for understanding. We could practice reading the dense writing and filling in the induction proofs that are "left for the reader". It would also be an occasion to remind the students of concepts learned in important courses in their major (e.g. modular arithmetic in an algebra course, or the topology of the real number line in an analysis course). And by the end of it, I hope the students would gain some satisfaction from realizing that they read and understood the entirety of a scholarly article in mathematics.

I am looking for other papers that could fill this role. Ideally, I think it would be great to construct a semester-long course based on reading, say, 5-10 papers like this and spending a week or two on each one. Each paper should:

1. be relatively short, say less than 6 pages (but obviously the density of writing plays a big factor). Ideally, we should be able to digest it the whole thing within a couple of weeks of careful reading.
2. be interesting and not too esoteric or specific. For instance, the result in the Cantor Set example above is surprising and interesting, and although I may have to remind students of what the Cantor Set is, I won't need to presume deep background knowledge in a specific topic. Or, Niven's proof that pi is irrational would be a good example. However, many of the "Proofs without words" in MAA journals, while perhaps fun, would not really give students practice with reading scholarly writing and the results may be too specific to inspire their interest.
3. tie together, or remind students of, some knowledge from core courses in the undergraduate curriculum, like algebra, analysis, calculus, combinatorics, or probability.
4. be published in a book or journal. (The Cantor Set example above was a footnote in the book Chaos and Fractals: New Frontiers of Science, according to this reddit post.)

Meta comment: The only similar question I could find on this site is MO.88946 ("Readings for an honors liberal art math course"), but it focuses on books for a popular audience and not scholarly mathematics per se. Also, MathEducators tends to avoid "community wiki"-style questions, so I decided to post here.)

Answer 1

I think this is a delightful paper:

Hull, Thomas C. "Solving cubics with creases: The work of Beloch and Lill." The American Mathematical Monthly 118, no. 4 (2011): 307-315. (PDF download.)

It features a female mathematician, origami, the classic problems of trisecting an angle and doubling the cube, and a "forgotten" method for finding roots of polynomials realized via "turtle geometry."

Abstract. Margharita P. Beloch was the first person, in 1936, to realize that origami (paper-folding) constructions can solve general cubic equations and thus are more powerful than straightedge and compass constructions. We present her proof. In doing this we use a delightful (and mostly forgotten?) geometric method due to Eduard Lill for finding the real roots of polynomial equations.

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Answer 2

I think Calkin and Wilf’s lovely short paper on what is now known as the Calkin-Wilf tree would be suitable.

Neil Calkin and Herbert S. Wilf, Recounting the Rationals. The American Mathematical Monthly, Vol. 107, No. 4. (2000), pp. 360-363

From the abstract: It is well known (indeed, as Paul Erdős might have said, every child knows) that the rationals are countable. However, the standard presentations of this fact do not give an explicit enumeration; rather they show how to construct an enumeration. In this note we will explicitly describe a sequence $b(n)$ with the property that every positive rational appears exactly once as $b(n)/b(n + 1)$.

Answer 3

I once found a pretty funny paper of Zagier and though about it as totally suited to this kind of purposes:

D. Zagier, How Often Should You Beat Your Kids, Mathematics Magazine, Vol. 63, No. 2 (Apr., 1990), pp. 89-92

It is four pages long and deals with elementary probability theory. The title is intriguing, and the game studied is as intuitive and common as possible. Yet, the answer is not so straightforward.

It has the extra appeal that it creates a dialectic with another article (the only one in the bibliography, also four pages long) that addresses a similar question. This shows pretty well how we should stay critical and do not stop with an answer, but keep investigating.

I think of it as a beautiful introduction to scholar papers.

Answer 4

Here is an interesting result published by Maxfield in 1970 [1] that I think would be suitable.

Given any positive integer $n,$ there exists a positive integer $N$ (indeed, for each $n$ there exist infinitely many such integers $N)$ such that the decimal digits of $N!$ begin with all the decimal digits of $n,$ in their correct order. For example, there exists an integer $N$ such that the decimal digits of $N!$ begin with the digits $314159265358979.$ According to Southard [2], the smallest values of $N$ for $n = 9$ and $n = 841$ are $N = 96$ and $N = 12745.$

To prove this result, Maxfield proves that the sequence whose terms are given by the fractional parts of $\log_{10} N!$ is dense in the interval $[0, 1],$ from which the desired result follows. Maxfield’s brief argument of how this desired result follows from the density condition was not entirely clear to me, but when I was reading through this paper about 10 years ago I managed to come up with a detailed proof for my personal notes.

Maxfield also proves (last 2 pages) that the sequence whose terms are given by the fractional parts of $\log_{10} \log_{10} N!!,$ and the sequence whose terms are given by the fractional parts of $\log_{10} \log_{10} \log_{10} N!!!,$ and so on (each finite iterate of the $\log_{10}$ function is evaluated at the same finite iterate of the factorial function) are each dense in the interval $[0, 1].$ Maxfield does not indicate any implications of this more general result, but one can show that it implies certain higher order decimal digit properties of iterated factorials. For example, note that each positive integer $m$ can be written as $10^{m^{*} \times 10^k},$ where $1 \leq m^{*} < 10$ and $k$ is an integer (e.g. $6 = 10^{7.778 \ldots \times 10^{-1}}$ and $2^{1000} = 10^{3.01 \ldots \times 10^2}).$ You can think of $m^{*}$ as a second order mantissa for the number $m.$ Now note that the fractional part of $\log_{10} \log_{10} m$ is equal to the fractional part of $\log_{10} m^{*},$ which is equal to $\log_{10} m^{*}.$ Maxfield’s density result for $N!!$ implies that for each positive integer $n,$ there exists a positive integer $N$ such that

$$ 10^{\log_{10} \log_{10} n} \leq 10^{\log_{10} \log_{10} (N!!)} < 10^{\log_{10} \log_{10} (n+1)} $$

It follows that given any positive integer $n,$ there exists an integer $N$ such that $n^{*} \leq (N!!)^{*} < (n+1)^{*},$ and hence the decimal digits of the number of trailing zeros of $N!!$ begin with all the decimal digits of $n,$ in their correct order. And there are higher order versions involving $N!!!$ and $N!!!!$ and so on.

Note: Although $N!$ ends with one or more trailing zeros for each $N \geq 5,$ not every positive integer can be the number of trailing zeros for $N!\,.$ For example, $24!$ has $4$ trailing zeros and $25!$ has $6$ trailing zeros, so $5$ is not a possible number of trailing zeros for $N!\,.$ Also, it is not difficult to see that the gaps can be arbitrarily large, as suggested by the fact that $(5^{300})!$ has $299$ more trailing zeros than $(5^{300} – 1)!\,.$

[1] John Edward Maxfield, A note on $N!$, Mathematics Magazine 43 #2 (March 1970), 64-67.

[2] Laura Southard, Investigations on Maxfields theorem, Pi Mu Epsilon Journal 7 #8 (Spring 1983), 493-495.

(ADDED NEXT DAY) What follows are some additional related results that I came across after I went through some of my things at home.

Theorems 1 and 3 on p. 74 of Diaconis [3] strengthen Maxfield’s result about the sequence of fractional parts of $\log_{10} N!$ by showing this sequence is actually uniformly distributed in $[0,1].$ Note that saying this sequence is dense in $[0,1]$ means that each subinterval (of positive length) of $[0,1]$ gets visited by some term (in fact, by infinitely many terms) of the sequence. On the other hand, saying this sequence is uniformly distributed in $[0,1]$ means that for each subinterval $I$ of $[0,1],$ the limiting relative frequency of the number of terms that visit $I$ (i.e. that belong to $I)$ is equal to length of $I.$ Intuitively, this means the terms of the sequence don’t accumulate at any given location faster than at any other location. Diaconis [3] is probably not suitable for careful reading by undergraduates (and it is longer than 6 pages), but the statements of some of its results and its discussions of related literature would be suitable.

A related result was proved in Moser/Macon [4], which is also a paper I think would be suitable for undergraduates to read. A summary of some of the results in [4] is given in my sci.math post cited below.

Although one might consider these “first digits” results as a recreational mathematics topic, they are related to Benford’s Law, something Diaconis [3] discusses. Regarding Benford’s Law, see A Bibliography of Publications about Benford’s Law, Heaps’ Law, and Zipf’s Law by Nelson H. F. Beebe and a google search for “Benford’s law” + “distribution of leading digits and uniform distribution”.

[3] Persi Warren Diaconis, The distribution of leading digits and uniform distribution mod 1, Annals of Probability 5 #1 (February 1977), 72-81.

[4] Leo Moser and Nathaniel Macon, On the distribution of first digits of powers, Scripta Mathematica 16 (1950), 290-292.

See also Abstract #5 on p. 520 in American Mathematical Monthly 58 #7 (August-September 1951) AND these papers AND my comments in this 26 September 2006 sci.math post.

Answer 5

I did read very carefully with a group of undergraduates the article PRIMES is in P by Manindra Agrawal, Neeraj Kayal, Nitin Saxena. It does not require anything above some modular arithmetic and some very basic knowledge of prime numbers, yet is a remarkable paper published in the Annals.

It gives the students the possibility to see some algebra and some arithmetic applied to a very important and old problem in mathematics. In my experience both groups of students enjoyed it a lot and, more importantly, learned a lot from the reading.

Answer 6

I just encountered this fun paper: "Half a coin": https://www.researchgate.net/publication/288609584_Half_of_a_coin_Negative_probabilities by Gabor J. Szekely. It asks the question "can we construct a 'half-coin', such that throwing it twice (independently) is the same as throwing a usual coin once"? It is short and accessible, using only mainstream techniques (and with the added twist that the linked pdf seems to be lacking one page, so some reconstruction is part of the fun).

And, as a public-relations campaign for Cross Validated https://stats.stackexchange.com/, let me mention a few posts there which could be used. Topic is probability:

https://stats.stackexchange.com/questions/333471/die-100-rolls-no-face-appearing-more-than-20-times/335132#335132 (A variant on the birthday problem)

https://stats.stackexchange.com/questions/41208/the-sleeping-beauty-paradox

https://stats.stackexchange.com/questions/211967/expected-number-of-times-to-roll-a-die-until-each-side-has-appeared-3-times (a variant of coupon collector problem)

Answer 7

One of the first papers I read as an undergraduate was Paul Erdős's Some remarks on the theory of graphs (3 pages). It has historical significance of one of the first papers in probabilistic combinatorics. (There are earlier papers but this is one of the most memorable.) The proofs aren't phrased in the modern terminology of the probabilistic method (for instance equation (1) carries around an extra factor of $2^{N(N - 1)/2}$ which disappears when phrased in terms of probabilities). This I think will help the students see mathematics as something that is being developed rather than something that was figured out centuries ago.

Answer 8

David R Richman, Some remarks on the number of solutions to the equation $f(X_1)+\cdots+f(X_n)=0$, Stud. Appl. Math. 71 (1984), no. 3, 263-266, MR0769080 (86d:11100) proves this theorem:

Let $S$ and $T$ be any finite sets, let $f$ be any function from $S$ to $T$, let $t$ be in $T$, and let $c_n$ be the number of solutions of $f(X_1)\#\cdots\#f(X_n)=t$, where # is any binary operation on $T$. Then the sequence $c_1,c_2,\dots$ satisfies a linear recurrence of degree at most $\vert T \vert$.

It's short, accessible, and remarkable to me that one can get any kind of conclusion from such general hypotheses.

Answer 9

In keeping with A Eremenko's comment, I suggest papers in the American Mathematical Monthly and, in particular, point to papers by Dan Velleman - who was the AMM Editor from 2007-2011, and writes very clearly. I suggest looking to any of his papers that have won awards; from his webpage:

Chauvenet Prize ("The fundamental theorem of algebra: A visual approach"), 2018

Chandler Davis Prize ("The fundamental theorem of algebra: A visual approach"), 2016

Paul R. Halmos - Lester R. Ford Award ("A drug-induced random walk"), 2015

Carl B. Allendoerfer Award ("Permutations and combination locks"), 1996

Lester R. Ford Award ("Versatile coins"), 1994

If pressed to suggest just one paper, then I would say the above-mentioned 2015 Halmos-Ford winner:

Velleman, D.J. (2014). A drug-induced random walk. The American Mathematical Monthly, 121(4), 299-317. PDF (no paywall).

As for his other papers, you could check google scholar. If pressed for one more suggestion, then I would point to one that is not mentioned above, which was co-written with Greg Call:

Call, G.S., & Velleman, D.J. (1993). Pascal's Matrices. The American Mathematical Monthly, 100(4), 372-376. JSTOR.

Answer 10

A solution of the Hilbert's third problem. The proof requires some knowledge in the linear algebra. In particular students need to know that $\mathbb{R}$ is a linear space over $\mathbb{Q}$. A detailed (and quite elementary) proof can be found in M. Aigner, G. M. Ziegler, Proofs from The Book. Fifth edition. Springer-Verlag, Berlin, 2014. You can find a lot of helpful resources in the Internet.

Answer 11

I recommend the following paper:

Monsky, P. (1970). On dividing a square into triangles. The American Mathematical Monthly, 77(2), 161-164.

Here are some expository pieces, after which I briefly mention why/how I believe it satisfies the criteria, and then I conclude with my own five point summary of the paper (which I think could be used as a way to frame this reading for students).

• Wikipedia

• What is Monsky's Theorem? by Moragues

• An Exposition of Monsky's Theorem by Sablan

• Algebraic Number Theory II (from page 109) by Clark

For the criteria specified above:

• be relatively short, say less than 6 pages (but obviously the density of writing plays a big factor). Ideally, we should be able to digest it the whole thing within a couple of weeks of careful reading.

Yes: it spans 4 pages, but the first and last are such small bits that it's about 2.5 pages in length.

• be interesting and not too esoteric or specific.

• tie together, or remind students of, some knowledge from core courses in the undergraduate curriculum, like algebra, analysis, calculus, combinatorics, or probability.

It's interesting, as it answers a question that can be asked in plain language:

If you divide a square into non-overlapping triangles such that each triangle has the same area, then must the total number of triangles be even?

(Any even number $2n$ of triangles is achievable: divide the square into $n$ equiareal rectangles and then divide each rectangle across a diagonal.)

The answer is yes: The total must be even i.e. this cannot be done with an odd number of triangles.

As to whether it is too esoteric: It connects combinatorics and valuation theory; the former is definitely not too esoteric (it's essentially Sperner's Lemma) and the latter hovers around number theory and ($p$-adic) analysis: so, it's unlikely that undergrads would have seen this part already, but the ideas should be graspable (I used it to introduce $2$-adic concepts to high school students!).

• be published in a book or journal

Yes, it's in the Monthly, which I link here. Alternatively, I have uploaded a copy here.

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Here is my own summary of the paper after reading it:

1. Classify unit square points with rational coordinates into three disjoint sets $A$, $B$, and $C$;

2. Use Sperner's Lemma to show there will be an $ABC$ triangle $T'$ for any divided square;

3. Show that translating $T'$ to a new triangle $T$ that sends the $A$ vertex to the origin doesn't change any of the vertex types;

4. If the two vertices not at the origin are $(x,y)$ and $(x',y')$, then we have:

$$\frac{1}{m} = \text{area }T = \frac{1}{2}|xy' - x'y| := \frac{1}{2} \cdot \frac{p}{q}$$

which means that $mp = 2q$ is even; the partial magic is setting this up so that $p$ is odd, after which we conclude it must be that $m$ is even.

5. The full magic is showing that everything "rational" above can be appropriately extended. This is not done in Monsky's paper (he references Lang's Algebra text) and I would recommend potentially leaving out these details for undergraduate students who haven't previously seen $p$-adic anything.

Alternatively, part five could provide one of many entry points for students who want to dive yet deeper into the mathematical content of this paper.

Answer 12

Something I wish more undergraduate time had been spent on is foundations and set theory, even if only some basic discussions, rather than just the passing comment or two (if that) in Algebra or Analysis. Here's a short paper which touches on a couple of related topics from set theory and complexity:

Martin Davis, The Incompleteness Theorem. Notices of the American Mathematical Society, Vol. 53, No. 4 (2006), pp. 414-418

The paper is only 5 pages long with few equations. It starts with some historical context, then discusses the incompleteness theorem. There are several opportunities to work out proof details not provided in the paper.

The second part of the paper discusses Peano Arithmetic and ZFC, touching on the arithmetic hierarchy. The last paragraph of the paper gives a proposition (without giving the proof) only provable in ZFC with a suitable large cardinal axiom.

There are a number of different directions you can take with this paper, hopefully it isn't too broad for your use.

Answer 13

Might László Lovász' resolution of the Kneser conjecture qualify?

Lovász, L. Kneser's conjecture, chromatic number, and homotopy. J. Combin. Theory Ser. A 25 (1978) 319–324.

Link

The statement of the conjecture needs only elementary combinatorics. The proof involves also elementary ideas of graph theory and algebraic topology, and a visually obvious theorem of Borsuk. Just under six pages.

I once heard the story that Lovasz submitted this originally for a higher doctorate, and then the whole document was only two pages. But this may well be a myth.

Answer 14

Depending on your students' backgrounds, the following might be appropriate.

There is Landau and Russell's simple proof of the Alon-Roichman theorem (6 pages) (this one requires some representation theory but could be reasonable depending on your students' background).

Landau, Zeph; Russell, Alexander, Random Cayley graphs are expanders: a simple proof of the Alon-Roichman theorem, Electron. J. Comb. 11, No. 1, Research paper R62, 6 p. (2004). ZBL1053.05060.

There is also A. Nilli's simple proof of the Alon-Boppana theorem (4 pages)

Nilli, A., On the second eigenvalue of a graph, Discrete Math. 91, No.2, 207-210 (1991). ZBL0771.05064.

Answer 15

I recommend the 7-page paper "College admissions and the stability of marriage" (pdf) by Gale and Shapley. Quoting from the paper's epilogue:

Most mathematicians at one time or another have probably found themselves in the position of trying to refute the notion that they are people with "a head for figures," or that they "know a lot of formulas." At such times it may be convenient to have an illustration at hand to show that mathematics need not be concerned with figures, either numerical or geometrical. For this purpose we recommend the statement and proof of our Theorem 1. The argument is carried out not in mathematical symbols but in ordinary English; there are no obscure or technical terms. Knowledge of calculus is not presupposed. In fact, one hardly needs to know how to count. Yet any mathematician will immediately recognize the argument as mathematical, while people without mathematical training will probably find difficulty in following the argument, though not because of unfamiliarity with the subject matter.

Theorem 1 of course is the famous Stable Marriage Theorem:

THEOREM 1. There always exists a stable set of marriages.

Answer 16

I propose Baumslag's paper

Gilbert Baumslag A non-cyclic one-relator group all of whose finite quotients are cyclic, J. Austral. Math. Soc., 10, 1969, 497–498.

This is where the group now known as the Baumslag group or Baumslag-Gersten group is defined.

The paper is a page long (the three references take up a second page) and elementary, and it whets the appetite for more - such as S. Gersten's proof that this group has a fast-growing Dehn function.

Answer 17

Another interesting read from the area of mathematical finance is the note from Yuri Kabanov and Christophe Stricker on the Fundamental Theorem of Asset Pricing (4 Pages):

Kabanov, Yuri, and Christophe Sticker. "A teacher’s note on no-arbitrage criteria." Séminaire de Probabilités XXXV. Springer, Berlin, Heidelberg, 2001. 149-152.

Link: A teacher’s note on no-arbitrage criteria

Answer 18

Why not Harris's beautiful (and IMHO underrated) masterpiece "Galois groups of enumerative problems"?

https://projecteuclid.org/euclid.dmj/1077313717

Also Selberg and Erdős's elementary proofs of the prime number theorem. With a patient yet prodding teacher at their fingertips it will benefit the students a lot to be able to wonder, what is the core essence of (various steps of the) proof? How do they point beyond themselves, i.e. why does this proof seem artificial in certain respects?

I gone through these in my undergraduate career with mentors, I found the experiences very helpful. The ideas that these papers point towards are also far more representative of what research mathematicians care about than say some of the other papers listed here.

Answer 19

I loved this:

On the rapid computation of various polylogarithmic constants

Authors: David Bailey, Peter Borwein and Simon Plouffe Journal: Math. Comp. 66 (1997), 903-913 MSC (1991): Primary 11A05, 11Y16, 68Q25 DOI: https://doi.org/10.1090/S0025-5718-97-00856-9 MathSciNet review: 1415794

It is about the Borwein-Bailey-Plouffe algorithm for directly computing the nth hexadecimal digit of pi and some other constants, without having to first compute all the preceding digits. I don't know if there is yet a corresponding formula to get pi's decimal digits.

Answer 20

Conway and Gordon's paper Knots and Links in Spatial Graphs proves two interesting elementary theorems:

• Any (non-pathological) embedding of the graph $K_6$ in 3D space contains two linked loops.
• Any embedding of the graph $K_7$ in 3D space contains a knotted Hamiltonian cycle.

The first theorem is proven in the first three pages and could stand on its own if constrained by time; the second theorem takes the next six pages, but they aren't very dense, although the appendix on the arf invariant isn't as nice (I think it's possible to translate most of its content into more elementary language though, at least if you only need the arf invariant mod 2).

Things you'd want to cover for an undergraduate audience reading the paper:

• The basic idea of knot theory (Reidemiester moves, the notion of a knot invariant)
• The idea of an ambient isotopy in $\mathbb{R}^3$ (I don't think you'd have to have topology as a prerequisite if you're willing to supply some visual intuition for the concept referred to here - the place they refer to this is making a quite visual argument.)
• The definition of a linking number, and the fact that it's a coherent well-defined property (probably comes just after an introduction to knot theory)