Exploring The Theory of Numbers with Pólya

One of the great challenges with any mathematics question is how to get started. George Pólya associates this with the general project of induction in science and in mathematics. He suggests that through engaging, practicing and comprehending the inductive process, we can become better mathematical problem solvers.

This is the first event in a series of three to explore several elementary but challenging questions in the theory of numbers. Participants are invited to accept the challenge to get started by joining us to inductively explore a bit of number theory.

Pythagoras' theorem relates the square (or second power) of the hypotenuse (the side opposite the right angle in a right triangle) to the sum of squares of the other two legs, in symbols, c²=a²+b². The search for pythagorean triples (https://en.wikipedia.org/wiki/Pythagorean_triple), triples of integers that form the lengths of the sides of a right triangle, may eventually suggest asking which numbers can be given by a sum of two squares? We quickly learn that not all integers can be so represented (3, for example, can only be composed by the sum of the squares 1²+1²+1² which requires three terms).

One of the more famous sum of squares problems was known to Diophantus of Alexandria (https://en.wikipedia.org/wiki/Diophantus) (c. 201-215-c. 285-299). It states that any positive integer can be represented as the sum of no more than four integers each of which is raised to the second power. Claude Gaspard Bachet de Méziriac (https://en.wikipedia.org/wiki/Claude_Gaspard_Bachet_de_M%C3%A9ziriac) (1581-1638) brought the problem to European attention when he translated Diophantus' Arithmetica (from the 3rd century CE). So it is sometimes called Bachet's conjecture. It was proved in 1770 by Joseph Louis Lagrange (https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange) (1736-1813) and so it is often called Lagrange's four-square theorem. Although we will explore the famous four-square theorem briefly, our effort will focus on other related questions about the sums of squares of integers.

This series of three events will discuss a short reading and a series of challenging problems related to the sums of squares in number theory. Our objective is to practice and thereby to more deeply understand the inductive process which is at the heart of getting started in exploring any mathematics question.

This topic will focus on a short 10 page passage (sections 4.1-4.7 on pp. 59-68 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n77/mode/2up)) and 6 example problems (examples #1-6 on pp. 70-71 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n89/mode/2up)) in Chapter 4 "Induction in the Theory of Numbers" in George Pólya's 1954 classic Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics (you can read the book for free on-line at https://archive.org/details/Induction_And_Analogy_In_Mathematics_1_ ; the "Read On-Line" format is excellent or you can download a PDF (https://archive.org/download/Induction_And_Analogy_In_Mathematics_1_/Induction_And_Analogy_In_Mathematics_1_.pdf) or if you click the "i" widget you can get copies in plain text, daisy, epub, or kindle formats).

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Although the reading for this session is light (only 10 pages). The example problems are challenging and will probably require substantial time and effort to solve (in particular, #4 and #5 look like they may require significant effort). Solving the example problems is not necessary to participate, but participants should spend at least a few hours trying to solve each of the 6 problems.

Preparation: Please prepare for the discussion with the following two steps:

• Read the 10 pages in the Chapter "Induction in the Theory of Numbers" sections 4.1-4.7 on pp. 59-68 which can be found through this link (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n77/mode/2up).

• Spend some time trying to solve the six (6) example problems #1-6 on pp. 70-71 which can be found through this link (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n89/mode/2up).

Note: since some of the problems are challenging, it is recommended that you plan to attempt the harder ones repeatedly over the course of several days. If you do get stuck on a problem, move on to the other problems (even ones that we will not discuss at this event): the problems are all related and one of them may give you some ideas about how to proceed. Also feel free to post questions to the event page so we can discuss what you are struggling with before the meetup.

To get a printable copy of the text, you may want to download the PDF of the whole book (https://archive.org/download/Induction_And_Analogy_In_Mathematics_1_/Induction_And_Analogy_In_Mathematics_1_.pdf).

Agenda.

• Detailed discussion of pp. 59-68 sections 4.1-4.6 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n77/mode/2up): what do you think of the motivation, the examples discussed, the method of tabulating and organizing calculations, the guessing of results? Do you understand each calculation, each sentence? What did you like about the text? What was unclear? Did you dislike anything about the text?

• Discussion about how to get started with a new mathematical question. Is the inductive approach exemplified in the text really the only way to get started on a mathematical question for which you do not have specific knowledge?

• Detailed discussion of your attempts at solving the six (6) example problems #1-6 on pp. 70-71 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n89/mode/2up). We will help anyone who got stuck and review your successes as well. Bring any questions you may have.

Interspersed with discussing the text and the problems or with any remaining time at the end we may also explore the following broader questions:

• What insights did working on these questions in the theory of numbers give into the project of Pólya's book to explore the principles of plausible reasoning and (scientific) induction (see sections 1-4 in the Preface (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n7/mode/2up) and Chapter 1 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n21/mode/2up))?

• In reading the text and working out the problems in Chapter 4 did you find supporting evidence for Pólya's claim on p. 7 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n25/mode/2up) that "A conjectural general statement becomes more credible if it is verified in a new particular case"? Did you find supporting evidence for his claim on p. 22 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n41/mode/2up) that "A conjecture becomes more credible by the verification of any new consequence"? Or his other claim on p. 22 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n41/mode/2up) "A conjecture becomes more credible if an analogous conjecture becomes more credible"?

• What if anything did you learn about the importance and the interplay of the roles of generalization, specialization and analogy (see Chapter 2 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n31/mode/2up)) in reading the text, working out the example problems, and/or reflecting on your efforts?

• What if anything did you learn about plausible reasoning by reading the text, working out the example problems, and/or reflecting on your efforts?

• What if anything did you learn about better understanding the art of problem-solving in reading the text, working out the example problems, and/or reflecting on your efforts?

• What if anything did you learn about the challenges in identifying relationships among integers and sums of their squares in reading the text, working out the example problems, and/or reflecting on your efforts?

• Do you think Bachet's conjecture (Lagrange's four-square theorem) is valid? Can you imagine how to prove it? Do you see a relationship between it and the other questions explored in the text and the example problems? Can you imagine how to get started to better understand the decomposition of integers into sums of squares and possibly re-discovering a proof of the Lagrange four-square theorem?

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Newcomers to Pólya's book can get very helpful but not required background by reading an additional 35 pages: the Preface (pp. v-x (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n7/mode/2up), pay particular attention to sections 1-4), the very important Hints to the Reader (pp. xi-xii (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n13/mode/2up)), Chapter 1 (pp. 3-11 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n13/mode/2up), pay special attention to the examples 9-14 on pp. 9-11 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n27/mode/2up)), Chapter 2 (pp. 12-30 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n31/mode/2up), pay special attention to examples 5, 7, 10, 11, 18-20, and 46). You should consider and compare the end of section 1.3 on p. 7 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n25/mode/2up) and section 2.7 on pp. 21-22 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n39/mode/2up). Reading this material will help you understand and assess Pólya's project to explore plausible reasoning, the nature of induction, and their relationship to mathematical problem-solving. Understanding the perspective of the book may help you in working on the example problems. In addition some of our discussion questions will explore this broader context of the book.

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This is the first event in a three part exploration of the theory of numbers with Pólya. The second event will be on 27 August "Inductive Discovery in Number Theory" (https://www.meetup.com/MathCounts/events/vwqvtlyvlbkc/). The final event in this series will be on 24 September "Exploring Integer Decompositions into Sums of Squares" (https://www.meetup.com/MathCounts/events/vwqvtlyvmbgc/).