Instructions
This problem set is due on Wednesday February 5, 2020 at 17:00. Solutions to Problems should be submitted on Gradescope and will be graded. Solutions to Exercises will not be graded and you should not submit them, although we will be happy to discuss them with you.
Problems
- Prove that if $p$ is prime, then $a^p = a \pmod p$ for all $a \in \mathbb Z$.
- Solve the congruence $x^{13} + 7x + 5 = 0 \pmod{91}$.
- Show that if $p$ is a prime and $p \geq 5$, then $p = \pm 1 \pmod 6$.
- Prove that if $n$ is a positive integer of the form $3m+2$ for a non-negative integer $m$, then $n$ has a prime factor of the same form.
Exercises
- Rosen 4.3, Exercises 13, 19, 23
- Rosen 4.3, Exercises 49, 55
- Rosen 4.4, Exercises 35, 37, 39
- Rosen 4.4, Exercises 65
- Rosen 4, Supplementary Exercises 29, 39, 41