Assignment 1, Math 313
Due: Friday, January 22nd, 2016
1 Let p and q be odd prime numbers with p = q + 4a for some a ∈ Z.
Prove that
a
a
=
.
p
q
2 If p is prime, gcd(p, 10) = 1, find a formula for
−5
p
.
3 If p and q are primes and a is an integer, what are the possible
numbers of solutions, modulo pq to the congruence
x2 ≡ a mod pq?
4 Does the congruence
x2 − 3x − 1 ≡ 0
mod 31957
have any solutions? Hint : 31957 is prime. Show your work (do not do
this by brute force!).
5 Show that there are infinitely many primes congruent to 2 modulo 3
by mimicking Euclid’s proof. Next, show that there are infinitely many
primes congruent to 1 modulo 3, by considering N = (2p1 p2 · · · pk )2 + 3
and using Legendre symbols.
6 Show that 15841 is an Euler pseudoprime to the base 2, a strong
pseudoprime to the base 2 and a Carmichael number.
7 Show that if n ≡ 5 mod 12 and n is an Euler pseudoprime to the
base 3. then n is a strong pseudoprime to the base 3.
1