prompter: How do I make a table in MathJax?

assistant: You can make a table in MathJax using the following syntax:
``` 
\begin{array}{column specification}
cell & cell & cell \\
cell & cell & cell \\
cell & cell & cell
\end{array}
``` 
The column specification defines the layout of the table, where each column is separated by & and each row is separated by \\. For example, the following code creates a simple 2x2 table with centered columns:
``` 
\begin{array}{c|c}
a & b \\
\hline
c & d
\end{array}
``` 
This produces the following table:
```
a|b
---
c|d 
```
You can adjust the column specification to create tables with different layouts and alignment options. For more information and examples, you can refer to the MathJax documentation on tables: https://docs.mathjax.org/en/latest/input/tex/arrays.html.

prompter: That link gives me a 404. Do you have another link that might work?

assistant: I apologize for the confusion. The MathJax documentation on tables seems to have been moved or removed. Here is an alternative resource that provides information on how to create tables in MathJax:

https://en.wikibooks.org/wiki/LaTeX/Tables#Using_the_array_environment

This resource provides an overview of the array environment in LaTeX, which is used to create tables in MathJax. The syntax is similar to what I provided earlier, but this resource includes additional information on how to specify column alignments and formatting options.

I hope this helps!

assistant: I apologize for the broken link. Here is the correct link to the MathJax documentation on tables: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference.

I hope this helps. Let me know if you have any other questions.

prompter: Can you express the same table in pure JavaScript?

assistant: One possible source of MathJax documentation on tables is a tutorial and quick reference page on Stack Exchange. It explains how to create arrays and tables with the array environment, how to align columns, how to add horizontal and vertical lines, and how to use multirow and multicolumn cells:

* https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference

Another possible source is the official MathJax documentation, which covers various aspects of using MathJax in web pages, such as installation, configuration, input formats, output formats, accessibility features, extensions, and more:

* http://docs.mathjax.org/en/latest/

* https://www.math.toronto.edu/MathJax/docs/html/index.html

assistant: To make a table in MathJax, you can use the LaTeX \begin{array}{} and \end{array} commands. Here is an example of a simple table with two columns and three rows:

```latex
$$
\begin{array}{|c|c|}
\hline
\text{Column 1} & \text{Column 2} \\
\hline
\text{Row 1, Column 1} & \text{Row 1, Column 2} \\
\hline
\text{Row 2, Column 1} & \text{Row 2, Column 2} \\
\hline
\text{Row 3, Column 1} & \text{Row 3, Column 2} \\
\hline
\end{array}
$$
```

This will produce a table with a border, two centered columns, and three rows. You can customize the table by changing the number of columns and rows, adding or removing borders, changing the alignment of the columns, and adding more complex formatting.

Note that you need to enclose the table code within double dollar signs ($$) to render it as a MathJax display equation.

Here's a breakdown of the code above:

\begin{array}{|c|c|}: This command starts the array environment and specifies the format of the table. | creates a vertical border, c centers the text in the column, and || creates a double vertical border between the columns.
\hline: This command creates a horizontal border.
Row 1, Column 1 and Row 1, Column 2: These are the entries in the first row of the table.
\end{array}: This command ends the array environment.
You can replace the text in the table with your own data, and adjust the formatting as needed.

I hope this helps!

assistant: To create a table in MathJax, you can use the following syntax:

\begin{array}{column1alignment column2alignment column3alignment ...}
  row1column1 & row1column2 & row1column3 & ... \\
  row2column1 & row2column2 & row2column3 & ... \\
  row3column1 & row3column2 & row3column3 & ... \\
  ... & ... & ... & ...
\end{array}

Here, column1alignment, column2alignment, column3alignment, etc. specify the alignment of the columns in the table. You can use l for left alignment, c for center alignment, and r for right alignment.

For example, here's how to create a simple table with two columns and three rows:

\begin{array}{c|c}
  x & f(x) \\
  \hline
  0 & 1 \\
  1 & 2 \\
  2 & 4
\end{array}

This will produce the following output:

\begin{array}{c|c}
  x & f(x) \\
  \hline
  0 & 1 \\
  1 & 2 \\
  2 & 4
\end{array}

You can adjust the alignment and the number of columns and rows as needed to create more complex tables.

prompter: Thanks for your answer, now I want that you solve the following problem: 

Let a and b be positive integers such that ab + 1 divides a² + b². Show that

a² + b² / ab + 1

is the square of an integer.

assistant: To show that a² + b² / ab + 1 is the square of an integer, we need to find an integer k such that:

a² + b² / ab + 1 = k²

To start, we can simplify the expression on the left-hand side by multiplying both the numerator and denominator by a - b:

(a² + b²)(a - b) / (ab + 1)(a - b) = k²

Expanding the numerator on the left-hand side and canceling out the common factor of a - b in both the numerator and denominator, we get:

(a³ - b³) + ab(a - b) / (a²b - ab²) + 1 = k²

Since ab + 1 divides a² + b², we can write:

a² + b² = n(ab + 1)

where n is some positive integer. Substituting this expression into the numerator of the left-hand side, we get:

(a³ - b³) + ab(a - b) = a³ - b³ + ab(a - b) = (a - b)(a² - ab + b²) = (a - b)n

Substituting this expression back into our original equation and simplifying, we get:

(a - b)n / ab - 1 = k²

Multiplying both sides by ab - 1, we get:

(a - b)n = k²(ab - 1)

Since a, b, and n are positive integers, we have:

a - b > 0 and ab - 1 > 0

Therefore, k² must also be a positive integer, and we can write:

k² = (a - b)n / (ab - 1)

Now we need to show that k is an integer. To do this, we will use the fact that ab + 1 divides a² + b². This can be written as:

a² + b² = m(ab + 1)

where m is some positive integer. Substituting this expression into the original equation, we get:

m(ab + 1) / ab + 1 = k²

Multiplying both sides by ab + 1, we get:

m(ab + 1) = k²(ab + 1)

Expanding the left-hand side and simplifying, we get:

mab + m = k²ab + k²

Dividing both sides by ab, we get:

m/a + m/b = k² + m/(ab)

Since m/(ab) is a positive integer and m/a and m/b are both positive integers, k² must also be a positive integer, and k is an integer.

Therefore, we have shown that a² + b² / ab + 1 is the square of an integer.