Generalization and Euler's Most Extraordinary Formula for the Sum of Divisors

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Generalization and its utility in making powerful guesses and arguments is the subject of chapter 6 entitled "A More General Statement" in George Pólya's 1954 classic "Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics" (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n109/mode/2up). At this event, we will deeply explore this chapter to better appreciate the use of generalization with examples from the Theory of Numbers.

The text features the translation of Leonhard Euler's memoir "Discovery of a Most Extraordinary Law of the Numbers Concerning the Sum of their Divisors" in §2 of chapter 6 on pages 90–98 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n109/mode/2up). Our workhorse for generalization will prove to be the deceptively simple idea of generating functions which Euler uses to great effect in his memoir.

To guide your preparation & participation, focus on the following questions and considerations which we will explore during the event:

• Can you give a clear explanation justifying each of the algebraic expressions on page 92 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n111/mode/2up). What caveats apply to using those formulas?

• On page 93, Euler explains his Most Extraordinary recurrence formula for σ(n) the sum of the divisors of n. How does it work? Compute σ(21), σ(22), and σ(23) using the formula to better understand its mechanics.

• In Euler's memoir, in the numbered sections 9-13 on pages 96-98 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n115/mode/2up), there is a sophisticated argument to justify his recursive formula for σ(n). Can you follow every step and explain each detail in that argument? Overall, what does the argument say? Is the argument persuasive? Is it a "perfect demonstration", that is, a proof? Why? Why not?

• In §3 and §4 of chapter 6 on pages[masked] (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n117/mode/2up), Pólya explains how generalization is used in Euler's memoir. What is the nature of generalization as used by Euler? What is the power of this approach to generalization? How persuasive is this kind of generalization? Are you convinced that Euler's Most Extraordinary law is true? Is the argument persuasive enough to justify Euler writing "which must be accepted as true although I am unable to prove it"? How can any argument, that is not a proof, be so persuasive? How could the argument fail? What is the difference between this kind of generalization and a "proper" proof? Are there multiple aspects of generalization at play here or is there only one kind of generalization?

• Address the problem posed in ex. 25 on page 107 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n125/mode/2up). What more did you learn about generalization in working through this example?

• The method of generating functions used in Euler's memoir is developed in chapter 6, ex. 1–8 on pages 101–102 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n119/mode/2up). If you are new to generating functions, it is highly recommended that you spend some time working on ex. 1-8 to get the basic idea.

• If any of the material is stumping you, please post a comment. We will try to help.

Pólya's book is in the public domain, so you can find free copies of it in PDF, EPUB, Kindle, text, and other formats at https://archive.org/details/Induction_And_Analogy_In_Mathematics_1_ (scroll down to the "Download Options").

In addition to the main material described above, the following passages introduce Pólya's approach in the book which will be useful for understanding the broader context of the material we are exploring but is not strictly necessary:

• The Preface on pages v–x (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n7/mode/2up), pay particular attention to §1–4.

• The very important Hints to the Reader on pages xi–xii (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n13/mode/2up).

• Chapter 1 on pages 3–11 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n21/mode/2up), pay special attention to ex. 9–14 on pages 9–11 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n27/mode/2up).

• Chapter 2 on pages 12–30 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n31/mode/2up), pay special attention to ex. 5, 7, 10, 11, and 18–20 on pages 23–30 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n41/mode/2up).