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David Pengelley

Publications and source records attributed to David Pengelley.

8 recordsLinked to original sources
arXiv2026

Sophie Germain, math\'ematicienne extraordinaire: A story stranger than fiction

Sophie Germain (1776-1831) was the first woman we know who did important original research in mathematics, specifically in elasticity theory and number theory. Celebrating her semiquincentennial year, we outline Germain's recently unearthed number theory results on Fermat's Last Theorem, in the context of her life, work, and interactions with Lagrange, Legendre, and Gauss. For two centuries her accomplishment on Fermat's Last Theorem was thought to consist of a single theorem attributed to her in a publication by Legendre, the first general result towards proving Fermat's Last Theorem. But recent discoveries in her handwritten manuscripts and correspondence with Legendre and Gauss show that she accomplished much more, albeit forgotten. In particular, she had a grand plan for proving Fermat's Last Theorem in its entirety, and carried this plan a long way, using then new tools, e.g., congruence, modular primitive roots, and permutations.

math.HO↗
arXiv2025

How did Fermat discover his theorem?

In 1640 Pierre de Fermat discovered his theorem that if $p$ is prime and $a$ is not divisible by $p$, then $a^{p-1}-1$ is divisible by $p$; or, as we write today, $a^{p-1}\equiv1\pmod{p}$. This is perhaps the first and the most important surprising property ever discovered about primes. There is little in number theory that is not dependent on it or intertwined with it, and its significance is amply demonstrated by the fact that today, almost four centuries later, Fermat's theorem provides the mathematical foundation for the RSA cryptosystem, which is still central to society's communications security even after several decades serving as its heart. Fermat's theorem is totally unexpected and truly astonishing. So why and how did he discover it? We know that Fermat was studying perfect numbers from classical Greek mathematics. But exactly how did that lead to his discovery? The secret lies in patterns in prime factorizations of Mersenne numbers, and Fermat's letters reveal hints of his path. We can reconstruct details of how Mersenne numbers led to Fermat's discoveries.

math.HO↗
arXiv2020

Towards active processes for teaching and learning

We discuss parallels between students and teachers in the process of pedagogical reform. Reform aims for students to develop their own process for becoming independent learners, and to gain personal ownership. Likewise teachers can develop their own personally owned reform process if they have the encouragement and freedom to take individual initiative. We argue that tools, such as text materials, technology, etc., are merely objects that should be kept in perspective as secondary within an overarching ever ongoing process. And we discuss how melding teacher freedom with collaboration can foster far-reaching change.

math.HO↗
arXiv2019

From lecture to active learning: Rewards for all, and is it really so difficult?

We describe the evolution of a personal non-lecture active learning pedagogy developed in numerous courses at all university levels. A distinguishing feature is its tight integration of pre-class preparation, involving student reading/writing/questions and problem work, with in-class active learning building on this preparation, and post-class follow-on homework. We discuss challenges, rewards, and buy-in for both students and instructors.

math.HO↗
arXiv2019

Evidence-based teaching: how do we all get there?

There are compelling reasons to shift our pedagogy toward evidence-based active learning methods that substantially improve student success, and now plenty of resources to aid in that shift. These include the recent CBMS Statement on Active Learning, MAA Instructional Practices Guide (IPG), and MIT Electronic Seminar on Mathematics Education. But implementation is neither quick nor easy. There are still plenty of individual, institutional, cultural, and professional obstacles, along with wonderful opportunities. At the 2019 Joint Mathematics Meetings we co-organized a guided discussion -- an ``un-panel'' -- sponsored by the American Mathematical Society's Committee on Education in order to stimulate the process of our community moving toward active learning in our teaching pedagogy. Seventy participants with fifteen discussion leaders expanded an initial list of issues, and considered questions around both challenges and opportunities. Here we summarize from these discussions, suggesting areas for collaborative efforts ranging from local colleagues and educational institutions to national and global professional societies.

math.HO↗
arXiv2016

How efficiently can one untangle a double-twist? Waving is believing!

It has long been known to mathematicians and physicists that while a full rotation in three-dimensional Euclidean space causes tangling, two rotations can be untangled. Formally, an untangling is a based nullhomotopy of the double-twist loop in the special orthogonal group of rotations. We study a particularly simple, geometrically defined untangling procedure, leading to new conclusions regarding the minimum possible complexity of untanglings. We animate and analyze how our untangling operates on frames in 3-space, and teach readers in a video how to wave the nullhomotopy with their hands.

math.GT↗
arXiv2010

"Voici ce que j'ai trouve": Sophie Germain's grand plan to prove Fermat's Last Theorem

A study of Sophie Germain's extensive manuscripts on Fermat's Last Theorem calls for a reassessment of her work in number theory. There is much in these manuscripts beyond the single theorem for Case 1 for which she is known from a published footnote by Legendre. Germain had a fully-fledged, highly developed, sophisticated plan of attack on Fermat's Last Theorem. The supporting algorithms she invented for this plan are based on ideas and results discovered independently only much later by others, and her methods are quite different from any of Legendre's. In addition to her program for proving Fermat's Last Theorem in its entirety, Germain also made major efforts at proofs for particular families of exponents. The isolation Germain worked in, due in substantial part to her difficult position as a woman, was perhaps sufficient that much of this extensive and impressive work may never have been studied and understood by anyone.

math.HO↗