Schedule

1. Understand mathematics as an experimental science using elementary number theory.
2. Learn mathematical proof methods to prove conjectures discovered during exploration.
3. Learn to use the SageMath software for computation and exploration.

• Emphasize that understanding mathematics as an experimental science (Goal 1) and learning proof methods (Goal 2) are more important than, but complemented by, using the computer algebra system SageMath (Goal 3).

• Problem Statement
  • Given coins of value a cents and b cents, determine what amounts of money can be obtained by combining them.
• Mathematical Formulation
  • Given positive integers \(a\) and \(b\), for which positive integers \(n\) do there exist non-negative integers \(x\) and \(y\) such that \(n = a\,x + b\,y\)?

• Begin with easy cases:\(\,a = 2, b = 3; a = 3, b = 4\).
• Solve the first two cases together to demonstrate the approach.
• Group students to explore more cases independently and later in groups.

• The conductor (\(N\)):the smallest integer \(N\) such that all \(n \geq N\) can be expressed as \(n = a\,x + b\,y\).
• The Frobenius number \(F\):the largest integer \(F\) such that no positive integers \(x\) and \(y\) exist with \(F = a\,x + b\,y\).

• Experiment with pairs of positive integers \(a\) and \(b\) to explore related questions.
• Encourage forming conjectures based on observations and use computers to explore complex examples (e.g., \(a = 19, b = 17\)).

• Use computers for more complex explorations.
• Introduce the SageMath software in the computer lab, with hands-on experience using a provided Jupyter notebook.

• Provided a brief overview of the main objective for the course. See Day 01 — Lecture Slides - 00
• Discussed the idea of mathematics as an experimental science using a problem statement.

• Solved two examples on the board
  • \(a=2\), \(b=3\)
  • \(a=3\), \(b=4\)
• Observed the Frobenius number and conductor for the pairs
  • For \((2, 3)\) Frobenius number = \(1\), Conductor = \(2\)
  • For \((3, 4)\) Frobenius number = \(5\), Conductor = \(6\)

• Noted some students struggled with understanding the definition of the conductor for a pair of positive integers.
• Wrote the definition of conductor on the board for clarity.

• Encouraged students to think of additional questions.
• Introduced a new question
  • Given two positive integers \(a\) and \(b\), where \(b\) is a multiple of \(a\), what are the Frobenius number and conductor?

• Students worked in groups to explore the posed questions and explored other questions as well.

• Conducted an introduction to the computer algebra system – SageMath – and guided students on how to start the software.
• Provided assistance to students as needed.
• Allowed students to work through the rest of the Jupyter notebook for the day and addressed questions asked by students to the whole class when necessary.

• Due on Saturday, 9th November, 2024 before 21:00.
• Assignment will be available on the Homework page on Thursday, 7th November, 2024.

• Begin doing daily assignments to prepare for the hand-in assignments.
• Emphasize the importance of completing these daily tasks.

• Given two positive integers \(a\) and \(b\) below, can you list all the positive integers \(n\) that cannot be written as \(n = a\,x + b\,y\)?
• \(a = 9\), \(b = 11\)
• \(a = 7\), \(b = 11\)
• \(a = 5\), \(b = 7\)
• Determine if there is a conductor for the sets \(\{2, 5\}\), \(\{3, 5\}\), and \(\{4, 6\}\). If so, identify it for each set.
• Explore if there is a Frobenius number for the sets \(\{2, 5\}\), \(\{3, 5\}\), and \(\{4, 6\}\). If so, identify it for each set.
• List all numbers that cannot be expressed as \(2\,x + 5\,y\), \(3\,x + 5\,y\), and \(4\,x + 6\,y\), and explore when this set is finite.
• Provide a step-by-step explanation of the conductor of \(\{3, 5\}\), including how to compute it and demonstrate that all \(N\) greater than the conductor can be expressed as \(3\,x + 4\,y\) for some positive integers \(x, y, \in \mathbb{Z}^{+}\).

• Get feedback from students on their first daily report and last lecture.
• Review the definitions of conductor and frobenius number.
• Work on the following steps on the cyclical process of experimental mathematics:
1. Start with a problem
2. Explore the problem by gathering enough data based of the posed problem.
3. Observe patterns within the data
4. Make intelligent conjecture(s)
5. Test the conjecture
6. Prove the conjecture
7. Generalize the conjecture
• Review notebook for Day 02

We gathered the data that students had collected thus far on the Frobenius number and conductor for certain values, but found it insufficient.

We then reviewed the notebook for Day 2, though we did not cover the exploratory questions in the notebook. I encouraged the students to formulate their own exploratory questions, resulting in the following list:

• If \(a\) and \(b\) share a common factor greater than \(1\), is there a Frobenius number and conductor?
• If \(a\) is even and \(b\) is odd, what are the Frobenius number and conductor?
• If \(a\) and \(b\) are not coprime, what are the Frobenius number and conductor?
• If the Frobenius number is infinite, is it always true that the conductor is also infinite, and vice versa?
• Is there a case where a Frobenius number exists but not a conductor?
• What are the conditions for the existence of a Frobenius number and conductor given \(a\) and \(b\)?
• If a Frobenius number and conductor exist, are they always unique?
• What happens if we consider more than two numbers, \(a\), \(b\), and \(c\)?
• Given that the Frobenius number and conductor exist, is there a relationship between the Frobenius number, conductor, \(a\), and \(b\)? If so, find it or identify the cases where such a relationship exists.
• If either \(a\) or bb equals \(1\) (but not both), does a Frobenius number and conductor exist?
• If both \(a\) and \(b\) are \(1\), what are the Frobenius number and conductor?

I encouraged students to initially explore these questions using pen and paper before transitioning to their computers. Although students were able to explore some of the questions in writing, most encountered significant challenges in translating their exploration into code. To address this, I organized a one-hour tutorial session in the evening, for which the notebook used is available in the resource section.

We will have a handed-in assignment (both computer and written) on Saturday 9th November, 2024. The problems to be handed-in will be on the Homework page on Thursday 7th November, 2024. The following problems will start getting you ready for it. You won't be able to do everything, but especially for 4 and 5, I strongly encourage you to start trying things and writing down what works and what doesn't work. Then for your assignment, you will really know how to get going.

1. Mathematical Proofs and Proof Techniques.
2. Find out how to do the following things in SageMath:
   1. Write an interesting function with variable \(x\)
   2. Take its derivative and integral
   3. Plot it for \(0 \leq x \leq 10\)
3. Find out the commands/methods for
   1. Making a diagonal matrix
   2. Finding the eigenvalues of a matrix
   3. Creating a new vector
   4. Taking the dot product of this vector with itself
   5. Finding the first \(100\) prime numbers
   6. Finding just the \(1000\)-th prime number
4. Prove that if \(a\) divides \(x\) and \(a\) divides \(y\), then \(a\) divides \(mx + ny\) for any integers \(m, n\).
5. Continue trying to explore the conductor. Here are more potential questions.
   1. How many numbers are not writeable as \(ax + by\) for the sets which have a conductor? Do you see a pattern in how many numbers are not possible?
   2. How does the problem change if instead of requiring nonnegative \(x, y\), we require positive \(x, y\) instead?
6. Create Sage code to do the following. Do not worry if you cannot get all of these, but you should start now to make as much progress as you can.
   1. List 10 numbers of the form \(4 x + 5\).
   2. List (at least) 20 numbers of the form \(4x + 5y\).
   3. List (at least) 20 numbers of the form \(4x + 5y\), but in numerical order.
   4. Write a function to list \(n\) numbers of the form \(4x + 5\), where \(n\) is some positive integer.
   5. Write a function to list (at least) \(n\) numbers of the form \(4x + 5y\), where \(n\) is a positive integer.
   6. Write a function to list numbers of the form \(5x + 6y\).
   7. Write a function with documentation that lists many numbers, in order, of the form \(ax + by\) for arbitrary \(a\), \(b\).

• Get feedback from students on last's night tutorial session and feedback report.
• Students should use dictionary to collect the data generated for each pair, \(a, b\).
• Remind students to take a look at the What to do for next time on Day 02 and instructions on homework page.
• Mathematical Proofs and Proof Techniques:
1. Proving implication statements using:
• Direct Proofs:
• Example 3.1.2 of restricted resource.
• Example 3.2.1 of restricted resource.
• Proof by Contrapositive
• Example 3.1.3 of restricted resource.
• Proof by Contradiction
• Example 3.2.2 of restricted resource.
• Example 3.2.3 of restricted resource.
2. Eliminating logical quantifiers and reduction techniques
• Students use the rest of the time to finish the notebooks for Day 01 and Day 02.

I gathered feedback from the students on their progress in the course, including their understanding of the material and overall well-being.

We discussed the use of dictionaries instead of lists, and students were encouraged to modify the code in the tutorial notebook to incorporate dictionaries.

We then proceeded to a discussion on mathematical proofs. The proof methods we covered included:

• Direct proofs
• Example 3.1.2 of restricted resource.
• Example 3.2.1 of restricted resource.
• Proof by contrapositive
• Example 3.1.3 of restricted resource.
• Proof by contradiction
• Example 3.2.2 of restricted resource.
• Example 3.2.3 of restricted resource.

For each method, I explained the underlying concepts and worked through examples with the students, using implication statements as a basis. Students were also given opportunities to construct proofs individually and present their solutions to the class.

I introduced the concept of a "design space" in the context of proving mathematical statements. This concept encourages students to identify their set of assumptions and determine precisely what needs to be proven. They can then use one of the proof strategies to reduce the problem to its essential components. We practiced this approach with examples, and I asked students to write formal proofs for some of these exercises, as demonstrated earlier on the board.

I reminded the class that writing formal proofs will be part of their homework assignment, due on Saturday, 9th November. Additionally, I distributed resources on mathematical proofs and techniques for their reference. I explained the strategy of eliminating the \(\forall\) quantifier and encouraged students to review the material on proof by mathematical induction, which was also included in the resources shared with them.

Find out how to do the following things in Sage:

1. Write an interesting function with variable \(x\)
2. Take its derivative and integral
3. Plot it for \(0 \leq x \leq 10\)

Find out the commands/methods for

1. Making a diagonal matrix
2. Finding the eigenvalues of a matrix
3. Creating a new vector
4. Taking the dot product of this vector with itself
5. Finding the first \(100\) prime numbers
6. Finding just the \(1000\)-th prime number

Continue trying to explore the conductor. Here are more potential questions.

1. How many numbers are not writeable as \(ax + by\) for the sets which have a conductor? Do you see a pattern in how many numbers are not possible?
2. How does the problem change if instead of requiring nonnegative \(x, y\), we require positive \(x, y\) instead?

Create SageMath code to do the following. Do not worry if you cannot get all of these, but you should start now to make as much progress as you can.

1. List 10 numbers of the form \(4 x + 5\)
2. List (at least) 20 numbers of the form \(4x + 5y\)
3. List (at least) 20 numbers of the form \(4x + 5y\), but in numerical order
4. Write a function to list \(n\) numbers of the form \(4x + 5\), where \(n\) is some positive integer
5. Write a function to list (at least) \(n\) numbers of the form \(4x + 5y\), where \(n\) is a positive integer
6. Write a function to list numbers of the form \(5x + 6y\)
7. Write a function with documentation that lists many numbers, in order, of the form \(ax + by\) for arbitrary \(a\), \(b\)

Suppose \(n\) is an integer greater than \(1\) and \(n\) is not prime. Then \(-1 + 2^n\) is not prime.

• Identify the hypothesis ("givens") and conclusion of the statement ("goal"").
• Is the hypothesis true when \(n = 6\)? What does the theorem tell you in this instance? Is it right?
• What can you conclude from the statement in the case \(n = 15\)? Check directly that this conclusion is correct.
• What can you conclude from the statement in the case \(n = 21\)? Check directly that this conclusion is correct.

Write a formal proof as I illustrated in class for the following statements:

1. Suppose \(A \subseteq C\), and \(B\) and \(C\) are disjoint. Prove that if \(x \in A\) then \(x \notin B\).
2. Suppose that \(A \setminus B\) is disjoint from \(C\) and \(x \in A\). Prove that if \(x \in C\) then \(x \in B\).
3. Prove that if \(a\) divides \(x\) and \(a\) divides \(y\), then \(a\) divides \(mx + ny\) for any integers \(m, n\)
4. Use the method of proof by contradiction to prove the following statements:
1. Suppose \(A \cap C \subseteq B\) and \(a \in C\). Prove that \(a \notin A \setminus B\).
2. Suppose that \(A \subseteq B\), \(a\in A\), and \(a \notin B \setminus C\). Prove that \(a \in C\).
5. Suppose that \(y + x = 2\,y - x\), and \(x\) and \(y\) are not both zero. Prove that \(y \neq 0\).
6. Prove that for every natural number \(n\in\mathbb{N}\), \(3\) divides \((-n + n^3)\).

The first handed-in assignment is due Saturday 9th November 2024 at 21:00! See the homework page.

Restricted resources on mathematical proofs and techniques sent by email to students.

Collect feedback from students on the previous night's tutorial session and feedback report.

Conduct a brief review of the prior day's lecture on mathematical proofs and techniques.

Engage in a discussion on proof by mathematical induction using Example 3.3.1 of the restricted resource.

Demonstrate to students how a formal proof is structured using the Example 3.1.2 of the restricted resource.

Explore the concept of divisibility and extended criteria related to the Frobenius number.

Reframe the Frobenius problem from a geometric perspective, as presented in the day's notebook.

The session began with a 20-minute period dedicated to gathering student feedback on the daily reports. It was collectively decided that students will present today's report solely on the lecture section covering proof by mathematical induction.

I provided students with a demonstration of a formal mathematical proof, using Example 3.1.2 from the restricted resource. Students were advised to adopt this style for writing proofs in their upcoming homework assignments.

I introduced the concept of a mini-tutorial on a subject of each student's choice and explained the @interact feature, which they will encounter in their second and third assignments.

We reviewed the day's notebook together, and students experimented with the @interact feature, which offers a geometric illustration of the Frobenius problem.

I addressed student questions regarding the implementation of the @interact feature.

The remaining lecture time was allocated for students to begin working on their homework assignments.

The first handed-in assignment is due Saturday 9th November 2024 at 21:00! See the homework page.

As preparation for tomorrow's class, you should also be exploring how to do calculus, linear algebra, differential equations, statistics, graph theory, number theory or some other area in SageMath; you will write your own mini-tutorial about this for the next handed-in homework assignment!

We will finalize our exploration on conductors by going to higher dimensions and watch this video that illustrates a practical application of the Frobenius problem.

I will introduce our new exploration question on integer sum of squares and collect possible questions from students that could be explored in the next week.

I concluded with remarks on the Frobenius problem, and we watched a video illustrating a practical application of this problem, specifically using the example of the number of chicken nuggets that cannot be ordered from McDonald's.

We reviewed the day's notebook, and I introduced a new problem on integer sums of squares. Students contributed potential questions to be explored in Monday's lecture. The notebook with these student-posed questions is available in the resource section, marked with an asterisk.

Students spent the remainder of the lecture working with SageMath. I noticed that several students were importing Python libraries like matplotlib, numpy, and math, so I reminded them that these packages are not required to be imported in SageMath.

I encouraged students not to be intimidated by errors, explaining that reading error messages is a valuable step in debugging their code.

I also demonstrated how to access the documentation for functions in SageMath, along with their source code. I emphasized that a useful approach to understanding how a function works is to read its documentation and to test the examples provided by copying them into a Jupyter notebook cell for further exploration to get a better understanding of the function.

I encouraged students to review the helpful links I provided on the resource pages, which should support their work with SageMath and inspire them to contribute to open-source software.

I concluded the lecture by reminding students of the upcoming homework assignment, due on Saturday, 9th November 2024 at 21:00. I also encouraged them to explore different areas of mathematics in SageMath for their mini-tutorial, which will be part of the next handed-in homework assignment.

The first handed-in assignment is due Saturday 9th November 2024 at 21:00! See the homework page.

As preparation for Monday's class, you should also be exploring how to do calculus, linear algebra, differential equations, statistics, graph theory, number theory or some other area in SageMath; you will write your own mini-tutorial about this for the next handed-in homework assignment!

Some Useful SageMath Resources:

• https://www.sagemath.org/sagebook/english.html
• https://www.sagemath.org/sagebook/french.html
• https://doc.sagemath.org
• https://www.sagemath.org/library.html
• https://www.sagemath.org/help.html
• https://www.sagemath.org/tour.html
• https://wiki.sagemath.org/Teaching_with_SAGE
• https://wiki.sagemath.org/Talks
• https://wiki.sagemath.org/quickref
• http://gregorybard.com/Sage.html

We will remind ourselves of the new exploration question on integer sum of squares and the possible exploratory questions we collected on Day 05, see the Jupyter Notebook.

We will divide ourselves into groups and collect enough data related to integer sum of squares in a tabular form on the board. Afterwards we will observe the patterns and try to make some conjectures if possible.

We will go to the computer lab and discuss an appropriate datatype to encode our tabular data and in SageMath. Aftwards, we will have ourselves in groups and start exploring the questions we collected on Day 05.

If the time permits, we will start plotting in SageMath and see how we can rethink the sum of squares problem geometrically.

The lecture began with the integer sum of squares problem. We formed small groups and, using pen and paper, calculated the sum of squares for all integers from 0 to 100. Students then recorded their results on the board in a table with three columns.

After examining the data, students proposed nine conjectures, which are now documented in the updated version of today's notebook.

We moved to the computer lab, where we discussed the optimal data structure for encoding the table we created. We determined that a **Python dictionary** would be the most suitable structure, organized as follows:

• Key: The sum of the squares, \( n = a^2 + b^2 \).
• Value: A list of two-element tuples, \([(a_1, b_1), \dots, (a_k, b_k)]\), such that \( n = a_i^2 + b_i^2 \) for \( 1 \leq i \leq k \), with \( k \geq 1 \).

While working on this, most students encountered challenges implementing the dictionary-based approach to collect and organize their data.

To help students test their conjectures, I demonstrated the following steps:

1. Translate Conditions: Convert the conditions stated in each conjecture into a Python function.
2. Filter Data: Use the Sage function filter together with the function created in Step 1 to filter the data and assess whether or not the conjecture holds. In some cases, additional functions were needed to validate or refute the conjecture.

This structured approach allowed students to explore the conjectures and deepen their understanding of mathematical concepts through programming.

Continue exploring the integer sum of squares problem and the conjectures proposed in today's lecture. Use the updated version of the Jupyter Notebook to guide your work.

Explore the questions we posed on Day 05 and consider how they might be addressed. See the updated Jupyter Notebook for day 5 for the questions.

Explore plotting in SageMath and see how we can rethink the sum of squares problem geometrically.

Some Useful SageMath Resources:

• https://www.sagemath.org/sagebook/english.html
• https://www.sagemath.org/sagebook/french.html
• https://doc.sagemath.org
• https://www.sagemath.org/library.html
• https://www.sagemath.org/help.html
• https://www.sagemath.org/tour.html
• https://wiki.sagemath.org/Teaching_with_SAGE
• https://wiki.sagemath.org/Talks
• https://wiki.sagemath.org/quickref
• http://gregorybard.com/Sage.html

We will begin by defining the congruence relation and applying it in our explorations.

I will encourage the students to consider the sum of squares from a geometric perspective, utilizing Sage's advanced plotting capabilities for visualization.

Our session will conclude with further investigation of the \(9\) conjectures developed in class, along with work on the remaining \(18\) exploratory questions.

I introduced the concept of the modulo arithmetic and demonstrated its application in Sage, particularly for exploring the sum of squares to facilitate pattern observation.

Students then worked individually to investigate the nine conjectures they formulated during Monday's class.

Since most students did not complete their exploration of all nine conjectures, I encouraged them to make time to finish, as this will be submitted as their daily report for today.

dditionally, I encouraged students to leverage Sage's advanced plotting features to explore the sum of squares problem from a geometric perspective.

Begin exploring techniques in SageMath for areas such as calculus, linear algebra, differential equations, statistics, graph theory, number theory, or other mathematical domains. For your next homework submission, you will create a mini-tutorial covering your chosen topic.

Additionally, continue working on the conjectures and exploratory questions provided in today's notebook

Some Useful SageMath Resources:

• https://www.sagemath.org/sagebook/english.html
• https://www.sagemath.org/sagebook/french.html
• https://doc.sagemath.org
• https://www.sagemath.org/library.html
• https://www.sagemath.org/help.html
• https://www.sagemath.org/tour.html
• https://wiki.sagemath.org/Teaching_with_SAGE
• https://wiki.sagemath.org/Talks
• https://wiki.sagemath.org/quickref
• http://gregorybard.com/Sage.html

While patterns can be helpful, today we will focus on the importance of not relying on them exclusively. Patterns may be necessary for forming conjectures, but they are not always reliable. Therefore, it is essential to develop skills in constructing and validating proofs.

We will organize into groups and work through today's notebook to practice reading and understanding mathematical proofs.

In the notebook for today, I introduce some concepts that may be new to some of you, offering further opportunities for exploration.

You will have ample time to explore problems related to the sum of squares. I encourage you to work in groups.

We began today's lecture by discussing the limitations of relying solely on patterns. Using the example of Fermat numbers—originally conjectured by Fermat to be prime—we illustrated how patterns can sometimes be misleading and emphasized the importance of constructing mathematical proofs.

Following this, I divided the students into groups to collaboratively work through the day's notebook. Most students progressed to the proof included in today's material. However, as no group was able to complete the section on the sum of squares, I encouraged them to finish this either independently or in group, noting that no daily report is required today.

I also reminded students to complete the 18 questions posed in last Friday's lecture. See Day 07 — Jupyter Notebook

Homework assignments will be posted on the course website this afternoon.

Begin exploring techniques in SageMath for areas such as calculus, linear algebra, differential equations, statistics, graph theory, number theory, or other mathematical domains. For your next homework submission, you will create a mini-tutorial covering your chosen topic.

Additionally, continue working on the conjectures and exploratory questions provided in today's notebook, and be sure to download the notebook for the second homework assignment.

Tomorrow, we will start a new experiment in the main lecture hall.

Some Useful SageMath Resources:

• https://www.sagemath.org/sagebook/english.html
• https://www.sagemath.org/sagebook/french.html
• https://doc.sagemath.org
• https://www.sagemath.org/library.html
• https://www.sagemath.org/help.html
• https://www.sagemath.org/tour.html
• https://wiki.sagemath.org/Teaching_with_SAGE
• https://wiki.sagemath.org/Talks
• https://wiki.sagemath.org/quickref
• http://gregorybard.com/Sage.html

Please note that the second homework assignment has been uploaded, with a deadline of Saturday, November 16, at 13:00.

Today, we will focus on the topic of integer partitions. I will start by defining integer partitions and introducing relevant terminology, followed by an example for clarity.

After the introduction, we will each work individually to compute the number of partitions of \(n\), where \(n \leq 10\).

Following this exercise, I will demonstrate how to formulate meaningful questions related to integer partitions, which will allow you to develop and explore similar questions on your own.

In the second hour, we will move to the computer lab for hands-on exploration of integer partitions.

Students were informed that the homework assignment has been uploaded, with a submission deadline of Saturday, November 16, at 13:00.

I introduced the concept of integer partitions, providing definitions and illustrating relevant terminology with an example.

Students then worked individually to compute the partitions of \(n\) for \(n \leq 10\), while examining potential patterns and forming conjectures. This exercise led to two conjectures and seven exploratory questions, which have been included in the updated version of today's notebook.

During the second hour, we moved to the computer lab to review the first section of the notebook. Students were introduced to the datatypes used within the Partitions class in Sage. The modified notebook, containing the questions students raised about integer partitions, was subsequently shared with them via email.

I also addressed all questions related to the second homework assignment.

Students worked through the notebook, focusing particularly on reading the documentation for the Partitions class and conducting exploratory exercises.

The lecture concluded with a reminder for students to explore the available Sage resources. In particular, the following resource is useful for working with integer partitions: SageMath Integer Partition Documentation.

We will continue our discussion on integer partitions, exploring additional questions, formulating conjectures, and working on some bijective proofs.

We will demonstrate Sage's @interact functionality. You will use this feature as a group to present a mathematical or scientific concept. For further details, refer to the project page.

Some Useful SageMath Resources:

• https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/partition.html
• https://www.sagemath.org/sagebook/english.html
• https://www.sagemath.org/sagebook/french.html
• https://doc.sagemath.org
• https://www.sagemath.org/library.html
• https://www.sagemath.org/help.html
• https://www.sagemath.org/tour.html
• https://wiki.sagemath.org/Teaching_with_SAGE
• https://wiki.sagemath.org/Talks
• https://wiki.sagemath.org/quickref
• http://gregorybard.com/Sage.html

Students reviewed the instructions and guidelines for the third assignment during class.

Students were divided into groups of at least \(4\) members to discuss their project ideas for the third assignment.

Each group read the @interact section of the notebook to familiarize themselves with its functionality.

The bijective proof technique was introduced and used to prove the conjecture:
\( p(n\,|\,\text{all parts even}) = p(n\,|\,\text{even number of each part}) \)

Groups collaboratively applied the bijective proof technique to prove the conjecture:
\( p(2n\,|\, \text{all parts even})=p(n) \)

Students were encouraged to begin working on the exploratory questions provided in today's notebook.

Students were instructed to read the guidelines provided on the projects page.

We organized into groups of at least \(4\) members to review the section of the notebook on @interact.

Group members engaged in discussions to brainstorm ideas for presenting a concept using Sage's @interact and animation features.

During the session, we observed that the layout of the @interact example provided in the Sage Interact Quickstart documentation did not function as expected.

We discussed the bijective proof technique and demonstrated its application by proving the statement:
\( p(n\,|\,\text{all parts even}) = p(n\,|\,\text{even number of each part}) \)

Students then applied the bijective proof technique independently to prove the statement:
\( p(2n\,|\, \text{all parts even})=p(n) \)

The second handed-in assignment is due Saturday 16th November 2024 at 13:00 GMT+2! See the homework page.

Continuing the brainstorming on project ideas for the third assignment using the @interact feature.

Continue working on the exploratory questions provided in today's notebook.

Some Useful SageMath Resources:

• https://www.sagemath.org/sagebook/english.html
• https://www.sagemath.org/sagebook/french.html
• https://doc.sagemath.org
• https://www.sagemath.org/library.html
• https://www.sagemath.org/help.html
• https://www.sagemath.org/tour.html
• https://wiki.sagemath.org/Teaching_with_SAGE
• https://wiki.sagemath.org/Talks
• https://wiki.sagemath.org/quickref
• http://gregorybard.com/Sage.html

Answer questions related to the @interact project.

Discuss arithmetic functions and their multiplicative properties.

Define two additional arithmetic functions:

• \(\tau(n)\):= The number of positive divisors of a positive integer \(n\).
• \(\sigma(n)\):= The sum of the positive divisors of a positive integer \(n\).

Prove that \(\tau(n)\) is a multiplicative function and discuss steps in the proof.

Implement \(\tau(n)\) and \(\sigma(n)\) in Sage.

Collect questions from students about \(\tau(n)\) and \(\sigma(n)\) for further exploration.

Informed students to practice their group presentations within the allocated time.

Proved that \(\tau(n)\) is multiplicative.

Proved that if \(p\) is prime, then \(\tau(p) = 2\) and \(\sigma(p) = p + 1\).

Discussed the Fundamental Theorem of Arithmetic.

Gathered questions about \(\tau(n)\) and \(\sigma(n)\) for further exploration using Sage.

Students explored the questions they had posed.

Prove Multiplicativity: Show that for all positive integers \(n\), the sum of divisors function, \(\sigma(n)\), is multiplicative.

Derive General Formulas: Derive a general formula for both \(\tau(n)\) and \(\sigma(n)\).

Explore Properties: Familiarize yourself with the properties of \(\tau(n)\) and \(\sigma(n)\) by exploring the questions provided in the updated version of today's notebook. Write a summary of your findings.

Some Useful SageMath Resources:

• https://www.sagemath.org/sagebook/english.html
• https://www.sagemath.org/sagebook/french.html
• https://doc.sagemath.org
• https://www.sagemath.org/library.html
• https://www.sagemath.org/help.html
• https://www.sagemath.org/tour.html
• https://wiki.sagemath.org/Teaching_with_SAGE
• https://wiki.sagemath.org/Talks
• https://wiki.sagemath.org/quickref
• http://gregorybard.com/Sage.html

Reminder:Students are reminded to review the instructions on the projects page.

Group Work: Form groups and work through today's notebook collaboratively.

Recap: Review yesterday's lecture and derive the general formulas for:

• The number of positive divisors function, \(\tau(n)\)
• The sum of positive divisors function, \(\sigma(n)\)

Encouragement: Motivate students to start utilizing Sage's @interact feature in their explorations, as demonstrated in the notebook.

Project Work: Dedicate the second hour of the lecture for students to work on their @interact projects.

I reminded students to review the instructions on the project page.

We divided into groups and collaborated on the contents of today's notebook.

I explained how to derive a general formula for the number of divisors function, \(\tau(n)\), and encouraged students to apply a similar approach to obtain the general formula for the sum of divisors function, \(\sigma(n)\).

I reminded students to utilize Sage's built-in function, sigma, for exploring the sum of divisors function, \(\sigma(n)\).

I demonstrated, with code, how to implement additional functions that can assist in our exploration.

Students worked on their Sage @interact group projects, with tutors and myself providing support as needed.

Continue working on your Sage @interact group projects.

Complete any remaining sections of today's notebook that were not finished in class.

Read the instruction on the projects page.

Some Useful SageMath Resources:

• https://www.sagemath.org/sagebook/english.html
• https://www.sagemath.org/sagebook/french.html
• https://doc.sagemath.org
• https://www.sagemath.org/library.html
• https://www.sagemath.org/help.html
• https://www.sagemath.org/tour.html
• https://wiki.sagemath.org/Teaching_with_SAGE
• https://wiki.sagemath.org/Talks
• https://wiki.sagemath.org/quickref
• http://gregorybard.com/Sage.html

Support students in completing their Sage interact group projects.

Groups that finish their work early work through today's notebook.

Students worked on their Sage @interact group projects.

No groups finished early, so we did not work through today's notebook.

Continue working on your Sage @interact group projects.

Submit your Jupyter notebook for the Sage @interact/animate group project by oday at 21:00 GMT+2. To submit, reply to the email I sent you with the notebook attached.

Read the instruction on the projects page as you prepare for your presentation today.

Some Useful SageMath Resources:

• https://www.sagemath.org/sagebook/english.html
• https://www.sagemath.org/sagebook/french.html
• https://doc.sagemath.org
• https://www.sagemath.org/library.html
• https://www.sagemath.org/help.html
• https://www.sagemath.org/tour.html
• https://wiki.sagemath.org/Teaching_with_SAGE
• https://wiki.sagemath.org/Talks
• https://wiki.sagemath.org/quickref
• http://gregorybard.com/Sage.html