Assignment 1

A common type of computation is the generate-and-test method, in which one systematically generates potential solutions to a problem, and then applies a sequence of one or more tests to determine if the proposed solution is in fact valid. While one could in principle (and under some circumstances one must) generate potential solutions randomly or according to some probability distribution, often it is more efficient to devise a systematic method for generating all candidate solutions. This is sometimes called solving a problem by "brute force".

Write a program that computes and prints the 1000th prime number. Time how long your program takes to run.

hint: the unix time program will time the execution of any command e.g. your program. So for example if your program is saved to a file called a01.py then you can time it like this:

time python a01.py

real    0m0.747s
user    0m0.695s
sys     0m0.011s

The line called "real" shows the total time taken to run the program, in this case \(747\) milliseconds. Time your program \(10\) times and report the average time taken to find the 1000th prime.

To help you get started, here is a rough outline of the stages you could follow in writing your code:

• Initialize some state variables
• Generate all (odd) integers \(>1\) as candidates to be prime
• For each candidate integer, test whether it is prime
• One easy way to do this is to test whether any other integer \(>1\) evenly divides the candidate with remainder \(0\). To do this, you can use modular arithmetic, for example, the Python expression a % b returns the remainder after dividing the integer \(a\) by the integer \(b\).
• You might think about which integers you need to check as divisors — certainly you don't need to go beyond the candidate you are checking, but how much sooner can you stop checking?
• If the candidate is prime, print out some information so you know where you are in the computation, and update the state variables
• Stop when you reach some appropriate end condition. In formulating this condition, don't forget that your program did not generate the first prime (\(2\)).

If you want to check that your code is correctly finding primes, you can go online and find a list of primes.

There is a cute result from number theory that states that for sufficiently large n the product of the primes less than n is less than or equal to \(e^{n}\) and that as \(n\) grows, this becomes a tight bound (that is, the ratio of the product of the primes to \(e^{n}\) gets close to \(1\) as \(n\) grows).

Computing a product of a large number of prime numbers can result in a very large number, which can potentially cause problems with our computation. (We will be talking about how computers deal with numbers a bit later in the term.) So we can convert the product of a set of primes into a sum of the logarithms of the primes by applying logarithms to both parts of this conjecture. In this case, the conjecture above reduces to the claim that the sum of the logarithms of all the primes less than \(n\) is less than \(n\), and that as \(n\) grows, the ratio of this sum to \(n\) gets close to \(1\).

To compute a logarithm, we can use a built in mathematical functions from Python. To have access to these functions, you need to get them into your environment, which you can do in Python by including this at the beginning of your program:

from math import *

This will allow you to use the function log() in your code, e.g. log(2) will return the logarithm base \(e\) of the number \(2\).

Write a program that computes the sum of the logarithms of all the primes from \(2\) to some number \(n\), and print out the sum of the logs of the primes, the number \(n\), and the ratio of these two quantities. Test this for different values of \(n\).

You should be able to make only some small changes to your program from above to solve this problem as well. While you should see the ratio of the sum of the logs of the primes to the value \(n\) slowly get closer to \(1\), it does not approach this limit monotonically.