Here are a series of tasks for you to attempt in Sage. You will want to reference “Mini Sage Introduction” and “Programming Basics” examples in the menu.
Task 1. Write a loop that will successively square a number modulo n. Use it to compute $2^{1024} \pmod {101}$. Make it show its work, by making informative print statements at each step.
Task 2. Create a while
loop which finds the first integer greater than 202 which is square. You can use the function is_square()
to check if something is a square. It returns True or False.
Task 3. Create a function that finds the first integer greater than $n$ which is a square, and returns it. You will want to adapt the code you just created above.
Here’s a separate box for testing your function.
Task 4. Create a program that computes the first 100 Fibonacci numbers and places them in a list.
Task 5. Create a function that implements Pollard’s p-1 factorization method, on any input integer $n$. You can use Sage’s inbuilt gcd.
Here’s a separate box for testing your function.
Task 6. Implement the Sieve of Eratosthenes on the primes from 1 to 100.
An extra box.