Exploring Integer Decompositions into Sums of Squares

Consider the Pythagorean theorem which 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. If we consider only integer solutions, we might wonder how to characterize all squares of integers which can be decomposed into the sum of two squares.

Eventually we might wonder what integers (not just which squares) can be formed 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). These and several related questions about the sums of squares of integers are elementary but challenging questions in the theory of numbers.

This may lead to the question of how many squares are needed to represent any positive integer? Or even how many solutions are there to the Diophantine equation n=x²+y²+z²+w² where n, x, y, z, and w are positive integers. Or perhaps where x, y, z, and w are positive odd integers. Or maybe let them be any integer? These problems are related to Fermat's theorem on the sum of two squares and Lagrange's four square theorem.

We will explore several questions related to number theory. A lot of our time will focus on several related problems that count the number of solutions to a Diophantine equation which decomposes integers into the sum of squares.

This topic will focus on a short 15 page passage (pp. 59-75 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n77/mode/2up)) and 11 example problems (examples #16-26 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n91/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 15 pages). The example problems are challenging and will probably require substantial time and effort to solve. Solving the example problems is not necessary to participate, but participants should spend at least a few hours trying to solve each of the 10 problems.

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

• Read the 15 page passage "Induction in the Theory of Numbers" on pp. 59-75 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 eleven (11) example problems #16-26 on p. 73 which can be found through this link (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n91/mode/2up). Although we try to minimize "prerequisites" for all our topics, it would be recommended to make sure you are familiar with examples #1-2 on pp. 70-71 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n89/mode/2up) and examples #9-15 on pp. 72-73 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n91/mode/2up) before you spend much effort on examples #16-23 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n91/mode/2up): those problems form a series that we split across three meetups (sorry it was too much for one meetup and we believe that there is an important lesson to learned from working all the problems).

Note: Although these problems are not hard (no advanced knowledge is necessary), they are very challenging. You will need to dedicate some effort to them.

Note: Not all of these problems involve the decomposition of numbers into the sums of squares. Example problems #24, 25, and 26 on p. 71 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n91/mode/2up) are at least not directly related to the sums of squares.

Note: Some participants have had difficulties understanding some of the problems (me included). Read them carefully. Please post a comment if you are not sure of your understanding of a problem. Some of the wording is a bit "loose" yet Pólya appears to have a particular idea in mind.

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: 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 anything 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 your attempts at solving the eleven (11) example problems #16-26 on p. 73 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n91/mode/2up). We will help anyone who got stuck and review your successes as well. Bring any questions you may have.

• Detailed discussion of pp. 68-70 sections 4.7-4.8 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n87/mode/2up) about the nature of inductive discovery and inductive evidence.

• Overview discussion of the whole of Chapter 4 on pp. 59-75 sections 4.1-4.8 plus all example problems #1-26 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n77/mode/2up). What do you think of the text, the examples, and the exercises? What did you learn? What unresolved questions or issues do you have with the text, its examples, or the problems?

Interspersed with discussing the text and the problems or with any remaining time at the end we will 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?

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Newcomers to Pólya's book can get some useful 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 final event in a three part exploration of the theory of numbers with Pólya. The first event was on 23 July "Exploring The Theory of Numbers with Pólya" (https://www.meetup.com/MathCounts/events/vwqvtlyvkbfc/). The second event was on 27 August "Inductive Discovery in Number Theory" (https://www.meetup.com/MathCounts/events/vwqvtlyvlbkc/).