Mathematics 153
Notes from April 29, 2020, lecture session

What was covered:
The material in the main file for Unit 5 up to the discussion of Pappus, augmented by solutions to two problems from the work of Diophantus:   One was the first problem in the file  http://math.ucr.edu/~res/math153-2020/week5/unit05/diophantus-examples.pdf   and the other was the first problem on the second page of the file http://math.ucr.edu/~res/math153-2020/week5/unit05/history05a.pdf.   More will be said about both problems in the next paragraph.  There was also a reminder about the short answer files posted to the main directory; these provide examples of the sorts of historical questions that will appear on the midterm examination.  Further information on the latter is discussed in another section of this document.
Since there is no recording of the April 29 lecture, here are a few more details on the coverage of new material.   The unit splits naturally into three pieces:  The first is a discussion of the work of Diophantus, the second is a discussion of the work of Pappus, and the third is a discussion of the last few developments in Greek mathematics.  Each of the first two is a source of mathematical questions, but the third is entirely historical.  
Two major features in Diophantus' work are (1) his departure from the earlier practice of formulating everything in words and developing his own shorthand (syncopated) notation for manipulating numbers, (2) his consideration of indeterminate algebraic problems which generally have many solutions over the positive rational numbers, although he generally was content to find a single example satisfying the conditions he imposed.    The full  impact of his work did not occur until well over 1000 years later when others studied some of his work in detail.   However, there was essentially no further progress in moving from words to shorthand and symbols for centuries afterwards.  Although others had studied similar problems for centuries before him, the writings of Diophantus are particularly significant due to the wide range and sheer quantity of problems he considered.   Usually the key steps in his solutions were clever insights and skill in making choices as opposed to general techniques which apply to wide ranges of problems.   This can be seen in the exercises for the course; usually there are extensive hints which include suggestions for how to get started.
In present day mathematics, the term  Diophantine equations  means something slightly different from the equations which Diophantus studied; namely, the modern term refers to finding integer solutions to polynomial equations with integer coefficients.  Results from the middle third of the 20th century showed that there is no systematic and universal method for determining whether such equations, or systems of equations, have a solution (the general question is known as Hilbert's Sixth Problem).
Here are some general comments about the solutions to the three equations considered in the lecture (one is in the main file and the other two are listed above).  Many of the algebraic expressions in the problems will simplify considerably if one unknown is somehow expressed in terms of the other.  For example, if the equation contains unknowns x and y, it is often useful to express y  =  x + C where C is still unknown but can be used effectively to retrieve y.   A second feature which appears repeatedly is to restrict attention to cases where the unknowns satisfy some well chosen constraint.   For the example worked out in the main file, the constraint is that the difference between the two unknowns is constant.  A third feature is to use some well known examples as a key step.  This is used in the proof that the square of a rational number can be written as the sum of two squares of rational numbers, which of course is based upon the existence of Pythagorean triples.
Regarding the later impact of the work, the link between the last example and Fermat's Last Theorem is discussed at length in the main file.  In the lecture I mentioned that the nonexistence of integral solutions for  x3 + y3  =  z3  (aside from the easy examples where  xy  =  0) can be proved using the techniques developed in an undergraduate abstract algebra course like Math 171.  Here is a step by step reference:
http://lostmathlessons.blogspot.com/2015/07/fermats-last-theorem-proof-for-n-3.html
In a slightly different direction, the main file mentions the question of representing a positive integer as a sum of squares, culminating with the four square theorem first proved near the end of the 18th century by Lagrange (generally pronounced la-GRANZH in English).  
The final topic covered in detail during the lectures was an example of a Diophantine problem which apparently predated Diophantus by centuries; namely, the so-called Archimedean cattle problem, which was designed as a challenge to others.   This problem has many solutions, and the first explicit and correct one, which has 206545 digits, was found in the 20th century using computers.  
Remarks on the midterm exam:
The examination will be a take-home test to be posted on Friday, May 1, and due on Wednesday, May 6.  There will be no class on Monday, May 4, as an accommodation for the exam.   Here are some answers to questions about the exam.
The exam will cover Units 1 through 4.  It will consist of 5 questions, probably about 70 per cent mathematical problems and about 25 per cent  short answer questions.  There is no time limit, but it should take at most two hours to complete (no guarantees, however; Your Mileage May Vary).   The levels of the problems will be close to those of the strongly recommended exercises, and they might use any mathematics up to and including first year calculus.   Outside references such as course directory documents may be used, and you may discuss the problem with other students, but the writeup you  submit is required to be your own work.   Additional examples of questions are in the folder http://math.ucr.edu/~res/math153-2020/old-exams  (solutions are given in the old examinations).  
Submission of exams.   Please submit THREE copies of your exam by electronic mail, one  to each of the following addresses, by 11:59 P.M. on Wednesday, May 6, 2020:
rschultz@ucr.edu, mover002@ucr.edu, tpier002@ucr.edu
Each problem should be started on a separate sheet of paper for the sake of efficient grading; the teaching assistants and I will be doing this jointly.
Regarding electronic submission, smart phone photos can be difficult or impossible to read, so any electronic submission should be easily printable and legible on a standard computer printer (e.g. , in pdf format as opposed to something that might not be widely available).

Relevant directory files for Unit 5:
Here is a listing of files related to the April 29 lecture with comments on their significance.  These links are all clickable.  
http://math.ucr.edu/~res/math153-2020/week5/unit05/history05.pdf  This is the main file for the subunit and should ultimately be read from beginning to end.  After doing so one should read the cited sections of Burton.
http://math.ucr.edu/~res/math153-2020/week5/unit05/diophantus-examples.pdf  This file gives solutions to some of the problems from Diophantus in the course text, and on the final page it illustrates the use of Pappus' Volume Theorem to find the volume for a specific solid of revolution (not covered in the April 29 lecture).  You should study these and understand them well enough to work similar exercises, in each case following the hints for the exercise.
http://math.ucr.edu/~res/math153-2020/week4/unit05/history05a.pdf   This file contains more detailed solutions to some problems worked out in the course text.
http://math.ucr.edu/~res/math153-2020/week4/unit05/history05b.pdf   AND
http://math.ucr.edu/~res/math153-2020/week4/unit05/history05c.pdf   AND
http://math.ucr.edu/~res/math153-2020/week4/unit05/history05d.pdf   
These three files consider some problems in Diophantus which are more complicated than the ones treated in the lectures.  You should read these to get a passive understanding of the methods.  None of this will be tested.
More files for Unit 5 will be listed in the summary for May 1.