# Unsolvable Math Problems Homework Hotline

As a 10-year-old boy in the 1960s, Andrew Wiles happened across a book in his local library called *The Last Problem*, which detailed a ~330-year-old struggle to solve the most longstanding unsolved problem in the history of mathematics: Fermat's Last Theorem.

Decades later, the now *Sir* Andrew Wiles – a professor of mathematics at the University of Oxford in the UK – has been awarded the prestigious Abel Prize for 2016, likened to the Nobel Prize of the math world. The award, bestowed by the Norwegian Academy of Science and Letters, carries with it a cash prize worth more than US$700,000, which some might say isn't such an extravagant reward for a proof described as "an epochal moment for mathematics".

Once he laid eyes upon Fermat's Last Theorem, the young Wiles was hooked on solving it, although he could never have guessed that the challenge would occupy the next three decades of his life.

"This problem captivated me," Wiles told Ian Sample at *The Guardian*. "It was the most famous popular problem in mathematics, although I didn't know that at the time. What amazed me was that there were some unsolved problems that someone who was 10 years old could understand and even try. And I tried it throughout my teenage years. When I first went to college I thought I had a proof, but it turned out to be wrong."

Put simply, the theorem, formulated by French mathematician Pierre de Fermat in 1637, states: "There are no whole number solutions to the equation x^{n}+y^{n}=z^{n} when n is greater than 2."

While the theorem can be expressed in such simple terms, solving it vexed mathematicians for some 350 years before Wiles' first proof was delivered in 1993.

That original solution – taking some 200 pages to write down – was the result of an intense period of research lasting seven years, during which Wiles lectured at Princeton University. When he delivered the proof in a series of lectures at Cambridge University, a crowd of some 200 researchers in attendance erupted in applause.

But even then, Fermat wasn't done. A mathematician reviewing Wiles' original work noticed errors in the solution, requiring the proof to be revised.

The final version was published in 1995 with the help of one of Wiles' former students, and the story behind the century-spanning solution generated such interest in the mathematics world (and outside of it) that a book on the saga became an international bestseller.

So how did Wiles solve what others couldn't for hundreds of years? By approaching the problem from an unconventional angle, combining elements of three branches of mathematics – modular forms, elliptic curves, and Galois representations – and building upon the work of centuries of mathematicians before him. Want a wee bit more detail? See here.

Now, with Wiles' latest recognition (he's already won several other awards), it's a fitting end to a race that began centuries ago, when a bold Fermat himself epically teased a solution to the theorem, before claiming that he didn't have enough space in his notes to write it down. "I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain," he wrote. (Fermat's own 'solution' was never found.)

For his part, Wiles says he hopes his efforts will encourage the next generation of curious 10-year-olds to dive into the challenges that mathematics offers.

"It is a tremendous honour to receive the Abel Prize and to join the previous Laureates who have made such outstanding contributions to the field," he said in a statement to the press. "Fermat's equation was my passion from an early age, and solving it gave me an overwhelming sense of fulfilment. It has always been my hope that my solution of this age-old problem would inspire many young people to take up mathematics and to work on the many challenges of this beautiful and fascinating subject."

*By Benjamin Braun, **Editor-in-Chief**, University of Kentucky*

One of my favorite assignments for students in undergraduate mathematics courses is to have them work on unsolved math problems. An unsolved math problem, also known to mathematicians as an “open” problem, is a problem that no one on earth knows how to solve. My favorite unsolved problems for students are simply stated ones that can be easily understood. In this post, I’ll share three such problems that I have used in my classes and discuss their impact on my students.

**Unsolved Problems**

*The Collatz Conjecture*. Given a positive integer \(n\), if it is odd then calculate \(3n+1\). If it is even, calculate \(n/2\). Repeat this process with the resulting value. For example, if you begin with \(1\), then you obtain the sequence \[ 1,4,2,1,4,2,1,4,2,1,\ldots \] which will repeat forever in this way. If you start with a \(5\), then you obtain the sequence \(5,16,8,4,2,1,\ldots\), and now find yourself in the previous case. The unsolved question about this process is: If you start from any positive integer, does this process always end by cycling through \(1,4,2,1,4,2,1,\ldots\)? Mathematicians believe that the answer is yes, though no one knows how to prove it. This conjecture is known as the Collatz Conjecture (among many other names), since it was first asked in 1937 by Lothar Collatz.

*The Erd*ő*s-Strauss Conjecture*. A fascinating question about unit fractions is the following: For every positive integer \(n\) greater than or equal to \(2\), can you write \(\frac{4}{n}\) as a sum of three positive unit fractions? For example, for \(n=3\), we can write \[\frac{4}{3}=\frac{1}{1}+\frac{1}{6}+\frac{1}{6} \, . \] For \(n=5\), we can write \[ \frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20} \] or \[\frac{4}{5}=\frac{1}{2}+\frac{1}{5}+\frac{1}{10} \, . \] In other words, if \(n\geq 2\) can you always solve the equation \[ \frac{4}{n}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\] using positive integers \(a\), \(b\), and \(c\)? Again, most mathematicians believe that the answer to this question is yes, but a proof remains elusive. This question was first asked by Paul Erdős and Ernst Strauss in 1948, hence its name, and mathematicians have been working hard on it ever since.

*Lagarias’s Elementary Version of the Riemann Hypothesis*. For a positive integer \(n\), let \(\sigma(n)\) denote the sum of the positive integers that divide \(n\). For example, \(\sigma(4)=1+2+4=7\), and \(\sigma(6)=1+2+3+6=12\). Let \(H_n\) denote the \(n\)-th harmonic number, i.e. \[ H_n=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} \, .\] Our third unsolved problem is: Does the following inequality hold for all \(n\geq 1\)? \[ \sigma(n)\leq H_n+\ln(H_n)e^{H_n} \] In 2002, Jeffrey Lagarias proved that this problem is equivalent to the Riemann Hypothesis, a famous question about the complex roots of the Riemann zeta function. Because it is equivalent to the Riemann Hypothesis, if you successfully answer it, then the Clay Mathematics Foundation will reward you with $1,000,000. While the statement of this problem is more complicated than the previous two, it doesn’t involve anything beyond natural logs and exponentials at a precalculus level.

**Impact on Students**

I’ve used all three of these problems, along with various others, as the focus of in-class group work and as homework problems in undergraduate mathematics courses such as College Geometry, Problem Solving for Teachers, and History of Mathematics. An example of a homework assignment I give based on the Riemann Hypothesis problem can be found at this link. When I use these problems for in-class work, I will typically pose the problem to the students without telling them it is unsolved, and then reveal the full truth after they have been working for fifteen minutes or so. By doing this, the students get to experience the shift in perspective that comes when what appears to be a simple problem in arithmetic suddenly becomes a near-impossibility.

Without fail, my undergraduate students, most of whom are majors in math, math education, engineering, or one of the natural sciences, are surprised that they can understand the statement of an unsolved math problem. Most of them are also shocked that problems as seemingly simple as the Collatz Conjecture or the Erdős-Strauss Conjecture are unsolved — the ideas involved in the statements of these problems are at an elementary-school level!

I have found that having students work on unsolved problems gets them engaged in three ways that are otherwise very difficult to obtain.

*Students are forced to depart from the “answer-getting” mentality of mathematics.*In my experience, (most) students in K-12 and postsecondary mathematics courses believe that all math problems have known answers, and that teachers can find the answer to every problem. As long as students believe this story, it is hard to motivate them to develop quality mathematical practices, as opposed to doing the minimum necessary to get the “right answer” sufficiently often. However, if they are asked to work on an unsolved problem, knowing that it is unsolved, then students are forced to find other ways to define success in their mathematical work. While getting buy-in on this idea is occasionally an issue, most of the time the students are immediately interested in the idea of an unsolved problem, especially a simply-stated one. The discussion of how to define success in mathematical investigation usually prompts quality discussions in class about the authentic nature of mathematical work; students often haven’t reflected on the fact that professional mathematicians and scientists spend most of their time thinking about how to solve problems that no one knows how to solve.

*Students are forced to redefine success in learning as making sense and increasing depth of understanding*. The first of the mathematical practice standards in the Common Core, which have been discussed in previous blog posts by the author and by Elise Lockwood and Eric Weber, is that students should make sense of problems and persevere in solving them. When faced with an unsolved problem, sense-making and perseverance must take center stage. In courses heavily populated by preservice teachers, I’ve used open problems as in-class group work in which students work on a problem and monitor which of the practice standards they are using. Since neither the students nor I expect that they will solve the problem at hand, they are able to really relax and focus on the process of mathematical investigation, without feeling pressure to complete the problem. One could even go so far as to evaluate student work on unsolved problems using the common core practice standards, though typically I evaluate such work based on maturity of investigation and clarity of exposition.

*Students are able to work in a context in which failure is completely normal.*In my experience, undergraduates majoring in the mathematical sciences typically carry a large amount of guilt and self-doubt regarding their perceived mathematical failures, whether or not it is justified. From data collected by the recent MAA Calculus Study, it appears that this is particularly harmful for women studying mathematics. Because working on unsolved problems forces success to be redefined, it also provides an opportunity to discuss the definition of failure, and the pervasive normality of small mistakes in the day-to-day lives of mathematicians and scientists. I usually combine work on unsolved problems with reading assignments and classroom discussions regarding developments in educational and social psychology, such as Carol Dweck’s work on mindset, to help students develop a more reasonable set of expectations for their mathematical process.

One of the most interesting aspects of using unsolved problems in my classes has been to see how my students respond. I typically ask students to write a three-page reflective essay about their experience with the homework in the course, and almost all of the students talk about working on the open problems. Some of them describe feelings of relief and joy to have the opportunity to be as creative as they wish on a problem with no expectation of finding the right answer, while others describe feelings of frustration and immediate defeat in the face of a hopeless task. Either way, many students tell me that working on an unsolved problem is one of the noteworthy moments in the course. For this reason, as much as I enjoy witnessing mathematics develop and progress, I hope that some of my favorite problems remain tantalizingly unsolved for many years to come.

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