I am writing this for participants of the CIMPA Summer School 2023 in Manila but this post may be useful for you if you want to learn more about PARI/GP.

This page is currently work in progress so feel free to email me if you find typos and inaccuracies!

Basics

When a ? and a rectangle appears, that means it is time to write some code!

Let’s start by breaking PARI.

Divide by Zero

Enter the following code

1/0

You will see an error because you’re not supposed to divide by 0.

Instead of the ? you will have break >.

Simply press Ctrl + D to go back to the normal ? prompt.

Long Loop

Now, we enter a function that does not finish for a long time:

for( i=1, 10^100, 1+1)

This is a loop that computes 1+1 for a total of 10^100 times.

Once this is running, you might want to stop or pause the computation. You can pause the computation by pressing Ctrl + C.

You will have the break > prompt again. From there you can either enter nothing and let the computation continue or press Ctrl + D to completely cancel the computation.

The Usual Suspects

Addition, subtraction, multiplication and division can be done using the usual +, -, * and / operators.

Parentheses can be used as well to group operations. Exponentiation is written as ^ and factorials are written as !.

For simplicity we will use x, y, and z for the variables of polynomials.

What polynomial does this code represent?

(x^2 - 1)*(x^2 + 2023!/2023!)

You can use a%b to get the remainder of a when divided by b when a and b are valid inputs.

Keeping Score

Whenever you write code that returns something (for example 1+1), PARI responds by giving something of the form:

%1 = 2

The number after the % indicates that it is the $1\text{st}$ output. And the = 2 means that the first output is 2.

TRY IT OUT: What does % return?

Exercise: Percent

If I write this and press ENTER:

5 + 6

What will be the output if I write this immediately after and then press ENTER?

%%3

Variables

The %n syntax is helpful for quickly accessing past computations. But you can assign variables.

For example, writing

two = 5
three = 7
3*two + 2*three

will give you 29.

Make sure to name your variables in a non-confusing way, unlike the example above.

For our purposes, NEVER assign x, y or z to any number unless you know how to “unassign” it. We will be using x, y, and z for variables in polynomials.

Pi

Pi is a variable which already has a value.

Type Pi and press Enter to see what PARI thinks its value is.

Space

Like an overly attached partner, PARI does not understand the meaning of space.

1            + 1

will give you the same result as

1+1

Various Other Tricks

If you press the Up and Down arrow keys, you can scroll through the previous commands that you’ve written. Useful if you need to modify a command that you just entered!

If you want to search for functions but have no idea what its name is, type ???"phrase" where you replace phrase with some words related to your function. For example, ???"number field" shows you functions that are associated to number fields.

If you have a function that you want to get to know further, type ??functionname where functionname is of course the name of the function. This will show the full documentation of that function — either as a PDF or on your terminal screen (depending on your computer).

Say you want to recap a function that you’ve used before and just want a summary — then you can use ?functionname. It will return a short summary that tells you how to use the function.

On some computers, you can also use Tab to give suggestions on how to complete what you’re writing. For example, typing nf and double pressing Tab gives you all the functions which start with nf.

Vectors

Constructing Vectors

Vectors can be constructed in PARI by using square brackets:

[1, 2, 3, 4, 5]

is a vector.

Example: Divisors Function

The divisors function returns a vector of the divisors of the input in ascending order.

For example:

divisors(2023)

returns

[1, 7, 17, 119, 289, 2023]

Example: Number field discriminant

Number fields are represented in PARI by a vector containing relevant information.

To get a representation of the number field

$K = \mathbb{Q}(z) = \mathbb{Q}[X]/(z^2+1)$

in PARI, we use nfinit:

K = nfinit(z^2 + 1)

The code above assigns to K a vector representing $K$.

According to the documentation, nfinit returns a vector whose 3rd component is the discriminant of the number field it represents.

To access the third component of the vector K, we use brackets:

K[3]

SURPRISE QUIZ: What would the above code return?

ONLY FOR PROS: Find a way to return the field discriminant without needing to remember its position in the vector.

Matrices

Example: Factor matrix

When given a positive integer $n$, the factor function gives a matrix whose first column contain the prime factors of $n$ and the second column contains their corresponding exponents.

For example

factor(5!)

gives you the matrix

[2 3]
[3 1]
[5 1]

showing that

$5! = 2^3 \cdot 3 \cdot 5.$

Accessing elements of a matrix

Just like vectors, you can access elements of a matrix by using brackets.

For example:

M = factor(5!)
M[3,1]

gives you 5, the entry on the third row and first column of the matrix M.

TRY IT OUT: What happens when you omit one of the two numbers? Say you put M[3,] or M[,1]?

GUESS AND CHECK: What do you think a ~ after a vector represents?

SURPRISE QUIZ: Make PARI return a vector containing the prime factors of $10^{50} + 2023$.

Writing Functions in PARI/GP

Example: Sum of Squares

In PARI, a function $f(x, y) = x^2 + y^2$ can be written as:

sumOfSquares(x, y) = {
    return(x^2 + y^2)
}

Hence,

sumOfSquares(3, 4)

will return 25.

Detour: Boolean expressions

In PARI (and in most languages), mathematical expressions involving inequalities get replaced with 1 if the statement is true and 0 otherwise.

And so expressions such as 2-1 < 3*2, 3 <= 2, and 1 > 2 will be either 0 or 1.

For asking whether two things are equal, we use ==. For example, 1 + 1 == 2 will return 1 (meaning true) but 1 + 1 = 2 will result in an error.

We can’t use only one = symbol because the = symbol already stands for “assign this expression on the right to the variable on the left”.

Detour: if function

In PARI, the if function returns the second argument if the first argument is true and returns the third argument otherwise.

SURPRISE QUIZ: What would the following code return?

if( 230-220 / 2 == 5! , 5 , 120)

TRY IT OUT: What would happen if you omit the second and third arguments in an if statement?

Example: isEven

Here is a function which takes as input a non-negative integer less than 4 and returns the integer 0 if it is odd and 1 if it is even.

isEven(n) = {
	if(n==0, return(1));
	if(n==1, return(0));
	if(n==2, return(1));
	if(n==3, return(0));
}

In the above function, we see the if function for the first time.

SURPRISE QUIZ: Find a partner and together write an improved isEven function, which works for any integer less than 10. Feel free to copy parts of the code!

SURPRISE QUIZ: Given two (horizontal) vectors x and y of equal length, write a function that returns their dot product.

Importing Files

Exercises

1. Write a function that, given an irreducible polynomial $f$ over the integers, computes the Minkowski bound of the number field defined by $f$.
2. Write a function that takes as input a negative integer $D$ and a rational prime $p$ and outputs whether $1$ if $p$ splits in $\mathbb{Q}(\sqrt{-|D|})$ and $0$ otherwise.