Math 480 (Spring 2007): Homework 5

Due: Monday, April 30

There are 3 problems. Each problem is worth 6 points and parts of multipart problems are worth equal amounts. You may work with other people and use a computer, unless otherwise stated. Acknowledge those who help you.

1. Encode the message NUMBER THEORY as a single number in base 27, where 0 corresponds to a space, A to 1, B to 2 and so on.

2. How many solutions does the following system of congruences have?
   

   $$x \equiv 3 \pmod{18}$$

   $$x \equiv 2 \pmod{3}$$

   $$x \equiv 1 \pmod{6}$$

   
   

3. In class I mentioned the famous open problem that there are infinitely many primes $p$ such that $(p-1)/2$ is also prime. Is it reasonable to conjecture that there are infinitely many primes $p$ such that $p\equiv 1\pmod{3}$ and $(p-1)/3$ is prime?