1.
Use the
next_prime() command to construct two different 8-digit prime numbers and save them in variables named a and b.Exercises 2.6 Sage Exercises
These exercises are about investigating basic properties of the integers, something we will frequently do when investigating groups. Use the editing capabilities of a Sage worksheet to annotate and explain your work.
2.
Use the
.is_prime() method to verify that your primes a and b are really prime.3.
Verify that \(1\) is the greatest common divisor of your two primes from the previous exercises.
4.
Find two integers that make a “linear combination” of your two primes equal to \(1\text{.}\) Include a verification of your result.
5.
Determine a factorization into powers of primes for \(c=4\,598\,037\,234\text{.}\)
6.
Write a compute cell that defines the same value of
c again, and then defines a candidate divisor of c named d. The third line of the cell should return True if and only if d is a divisor of c. Illustrate the use of your cell by testing your code with \(d=7\) and in a new copy of the cell, testing your code with \(d=11\text{.}\)
