Next: About this document ...
Homework Assignment 1
Due September 26, 2001
William Stein
Date: Math 124 HARVARD UNIVERSITY Fall 2001
Instructions: Please work in groups, and
acknowledge those you work with in your write up. Some of the problem below, such as
``factor a number'' can be quickly done with a computer. Feel free to
do so, unless otherwise stated.
 Let be a prime number and and integer such that
. Prove that divides the binomial
coefficient
You may not assume that this coefficient is a integer.
 Compute the following gcd's using a pencil and
the Euclidean algorithm:
 Using mathematical induction to prove that
then find a formula for
 What was the most recent prime year?
I.e., which of
was it?
 Use the Euclidean algorithm to find integers
such that
[I did not tell you how to do this; see §1.8 of Davenport's book.]
 Factor the year that you should graduate from Harvard
as a product of primes. E.g., frosh answer
.

Write a PARI program to print ``Hello Kitty'' five times.
 Let
be a polynomial with integer coefficients.
Formulate a conjecture about when the set
and $f(a)$ is prime
is infinite. Give computational evidence for your conjecture.
 Is it easy or hard for PARI to compute the gcd
of two random 2000digit numbers?
 Prove that there are infinitely many primes of the form .
 Use PARI to compute
primes
 The prime number theorem predicts that is
asymptotic to . How close is to
?
Next: About this document ...
William A Stein
20011011