Number Theory Through the Eyes of Sophie Germain

It takes tremendous erudition for an author to fully understand Number Theory Through the Eyes of Sophie Germain, and then lucidly explain it to others in a manner that is both captivating and instructive. David Pengelley has done just that in this unique book that will motivate many students to pursue research in this the most pure and enchanting branch of mathematics. To truly grasp the beauty thus exposed, this book demands effort, perseverance, and research on the part of the students and their teacher. It is a workbook that requires background in number theory, experience proving propositions, mathematical maturity to appreciate the analysis of theorems proven by remarkable mathematicians, and enthusiastic desire to tackle many exercises, some of which are based on famous conjectures (e.g. Goldbach’s conjecture).  

 

Upon receiving the copy of Pengelley’s newly published work, I was thrilled and intrigued. Knowing that Sophie Germain had left her most important mathematical legacy imprinted in a few pages of hand-written correspondence, I could appreciate the tremendous challenge for anyone attempting to understand her analysis and then translate it into modern mathematical form to present her unique approach to solving Fermat’s Last Theorem (FLT). As experts can attest, working with original sources requires that they place important and fundamental ideas in the context of the problem the author of the original work wished to solve. Of course, Pengelley is one of those experts, and he has studied with scholarly rigour Germain’s mathematics through her few available manuscripts. After poring through the first chapters, I was awed by the priestliness of the exposition where Pengelley highlights the beauty of Germain’s own attempts at proving one of the most important theorems in number theory, that only Euler before her had undertaken. Reading through, I began to learn how such endeavour [using primary sources] can be used to successfully teach mathematics and its history. 

 

As a prelude to presenting Germain’s work to prove FLT, the student is given nine exercises to warm up, differing in complexity to solve, requiring also familiarization with famous problems. For example, Exercise 2.8 asks the student to show that for any \( n>1 \), there is a prime between \( n \) and \( 2n \) for which the theorem of Bertrand/Chebyshev could be used. Exercise 2.9 requires to prove that, if Goldbach’s conjecture (still unproven) is true, then every odd number \( n \geq 7 \) is the sum of three odd primes (known as Goldbach odd conjecture or ternary Goldbach conjecture). 

 

In Number Theory Through the Eyes of Sophie Germain the student will encounter Euclid, Fermat, Euler, Lagrange, Legendre, and Gauss, discovering along the way many fascinating pieces of their work interwoven in a manner that facilitates learning and a deeper understanding. For example, after remarking that Euler proved most of Fermat’s assertions, Pengelley introduces Euler’s proof of FLT for exponent 4 and then prompts students to explain in their own words the method of descent based on Euler’s algorithm. This type of didactic presentation is necessary to fully grasp the essence of Germain’s own attempt at the general proof.  

 

Having been her mentor, the contributions of Legendre, the greatest number theorist in France, form a strong foundation for learning with this book, and especially as Germain communicates with him her most important results, some of which Legendre included in the Second Supplement of his Essai sur la théorie des nombres (1825) Of course, a study of number theory cannot end with the sources from where Sophie Germain learned her mathematics. Pengelley thus introduces us to the work of others after her. For example, to Eisenstein who proved the fundamental theorem of quadratic reciprocity using geometric arguments (a partial adaptation of Gauss’s third proof), and he remarks on the significance of Sophie Germain Primes even today. 

 

The most precious jewels of Number Theory Through the Eyes of Sophie Germain are contained in chapters 3, 8, 9, and 10 where Pengelley presents in detail how Germain carried out her “grand plan” to prove la célèbre équation de Fermat, as she called it. Pengelley dissects how Germain arrived at her own theorem, and he masterfully elucidates the meaning in Germain’s short letter to Legendre admitting that her plan did not lead to the ultimate proof she sought. Chapter 3 introduces the first letter Germain wrote to Gauss in 1819, remarking on the impossibility of satisfying in whole numbers the equation \( x^{p}+y^{p}=z^{p} \), as a prelude to outlining her ideas to prove Fermat’s theorem. This is a manuscript that requires careful consideration since she does not explain fully her arguments. Germain writes tersely because she is addressing her mathematical ideas to Gauss, one of the most remarkable masters in the field, a contemporary géomètre (as mathematicians were called then) she deeply admired and sought (through correspondence) for validation of her own work. Pengelley tasks the students to research, prove theorems, and show the validity of many assertions related to arithmetic progressions and power residues, including the non-consecutivity condition, important topics in Germain’s own work.

 

In chapter 10, Pengelley guides the student in a sequence of exercises to verify Germain’s mathematics, admitting that “the tasks will challenge you”; it is here where the material learned in previous chapters becomes critically important to understanding Germain, her arguments, and the complexity of the famous assertion that she singlehandedly  undertook, attempting to fully prove it.

 

Even in the appendix, where Pengelley describes how Fermat discovered his most famous theorem, the student is presented with 33 tasks and exercises of varied complexity involving Mersenne numbers, perfect numbers, and several of Fermat’s less-known assertions. There are problems such as “Find the first number of the form \( M_{p}=2^{p}-1 \) where p is prime, such that \( M_{p} \) itself is not a prime,” and “prove that no prime \( p \equiv 7 \mod 8 \) can be a divisor of any number of the form \( 2^{i}+1 \).”  These exercises will enrich, further deepening the understanding of Germain’s monumental endeavours, and strengthening the skills of the students to prove some of the most beautiful theorems in number theory.

 

Learning number theory from this type of book, I believe, places more challenging expectations on the students, as the material requires a strong foundation in mathematics to prove, verify, and investigate many aspects of the topics covered.  However, the intellectual rewards will be exponentially greater and much more satisfying; finishing a course with Pengelley’s book will place students on solid ground in preparation to becoming great number theorists.   

 

For adopting Number Theory Through the Eyes of Sophie Germain as a text to teach undergraduates with original sources, the instructor may need to begin with an overview of fundamental topics such as properties of prime numbers, infinitude theorem, Fundamental Theorem of Arithmetic, residues, congruences, congruences with a prime-power modulus, and especially to introduce the correct notation to tackle the proofs in this book.  Although many students could have encountered some of those topics already, the material may not have treated them as formally as required to fully appreciate and study Pengelley’s magnificent book.

---

 

Dora Musielak, University of Texas at Arlington