`factor(2021)`

How would you install PARI/GP on a Linux system?

First, you would download the source code from the PARI/GP website. Then, you would extract the files, navigate to the directory, and run './Configure' and 'make all' commands. Finally, run 'make install' to finalize the installation.

What are some of the functions provided by PARI/GP for number theory computations?

PARI/GP provides many functions for number theory computations, including prime number functions, functions for factorization, functions for handling modular forms, and functions for elliptic curves.

How would you use PARI/GP to compute the factorial of a number?

You can use the 'factorial' function in PARI/GP. For example, to compute the factorial of 5, you would enter 'factorial(5)' at the prompt.

What is the purpose of the 'ellinit' function in PARI/GP?

'ellinit' is used to initialize an elliptic curve. It takes as input the coefficients of the elliptic curve and returns an object representing the curve, which can then be used in further computations.

How would you define a function in PARI/GP?

Functions in PARI/GP are defined using the 'fun' keyword, followed by the function name, arguments in parentheses, and the function body enclosed in braces. For example, 'fun square(x) = { return x^2; }'.

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Does the candidate have a strong understanding of PARI/GP?

Has the candidate demonstrated problem-solving skills?

Is the candidate familiar with number theory and algebra?

Can the candidate work well in a team?

Describe the difference between the 'factorint' and 'factor' functions in PARI/GP.

Both 'factorint' and 'factor' functions perform factorization, but 'factorint' is used for factoring integers, while 'factor' can be used for factoring polynomials as well as integers.

What are some ways to improve the performance of PARI/GP scripts?

Performance can be improved by using vector operations instead of loops where possible, preallocating memory for large data structures, and using the inline version of functions when the overhead of a function call is significant.

How would you use PARI/GP to compute the sum of the first n terms of a geometric series?

You can use the 'sum' function in combination with the 'for' loop. For example, to compute the sum of the first n terms of the series 2^n, you would enter 'sum(n=1, N, 2^n)'.

What is the purpose of the 'forprime' function in PARI/GP?

'forprime' is a loop construct that iterates over all prime numbers in a given range. The body of the loop is executed for each prime number.

Describe the difference between the 'if' and 'ifelse' functions in PARI/GP.

'if' and 'ifelse' both perform conditional execution, but 'ifelse' is an expression that returns a value, while 'if' is a statement that does not.

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What does the following PARI/GP code do?

`factor(2021)`

This code gives the prime factorization of the number 2021.

What will be the output of the following PARI/GP code?

`a = vector(5, i, i^2); a`

This code will output the first five squares: [1, 4, 9, 16, 25].

What does the following PARI/GP code do?

`a = vector(5, i, i^2); vecsort(a, ,4)`

This code creates a vector of the first five squares and then sorts this vector in ascending order based on the 4th entry.

What will be the output of the following PARI/GP code?

`install(numtoperm, GG); numtoperm(5,3)`

This code will output the 3rd permutation of 5 elements, which is [1,3,2,4,5].

How would you use PARI/GP to compute the greatest common divisor of two numbers?

You can use the 'gcd' function in PARI/GP. For example, to compute the greatest common divisor of 12 and 18, you would enter 'gcd(12, 18)'.

What are the benefits of using PARI/GP for mathematical computation over other software?

PARI/GP has a number of benefits over other software, including its speed, its wide range of built-in functions for number theory, and the flexibility of its scripting language.

How would you implement the Sieve of Eratosthenes in PARI/GP?

You would create a list of numbers from 2 to the desired limit, then repeatedly remove multiples of the smallest remaining number, until no numbers are left. This can be implemented with a combination of 'for' loops and array operations.

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