University of Toronto
Faculty of Arts and Sciences
MAT315H1 - F: Introduction to Number Theory
Fall 2023
Homework 7
1 Problems to be submitted
• Make sure you follow all the indications as stated in the syllabus.
1. (5 points) In lecture we will prove that there is an infinite number of prime numbers of the forms 7k+2.
Using the same strategy, prove that there are an infinite number of primes of the form 5k+1, 5k+2, 5k+3
and 5k + 4.
Hint: You did the character table modulo 5 in homework 6.
2. (a) (3 points) Construct the character table modulo 12.
Hint: Use problem 3 of this homework. In lecture we already constructed the tables modulo 3 and
modulo 4.
(b) (2 points) Prove that there is an infinite number of primes of the form 12k + 5.
3. Let m and n be two positive integers that are relatively prime.
(a) (3 points) Let ϕ : Z→ C be a Dirichlet Character modulo m and ψ : Z→ C a Dirichlet Character
modulo n. Define χ : Z→ C by
χ(n) = ϕ(n)ψ(n).
Justify that χ is a Dirichlet Character modulo mn.
(b) (3 points) Let χ : Z → C be a Dirichlet Character modulo mn. Prove that there exist a Dirichlet
Character ϕ : Z → C modulo m and a Dirichlet Character ψ : Z → C modulo n such that χ is
constructed out of ϕ and ψ as in part (a).
Hint: To define ϕ at the value a, use the Chinese Remainder Theorem to construct an integer A
with A ≡ a (mod n) and A ≡ 1 (mod m), and solve for ϕ(a).
(c) (2 points) Justify that for a given Dirichlet Character χ : Z → C modulo mn, the ϕ and ψ con-
structed in the previous part are unique.
(d) (2 points) How many different Dirichlet Characters are there modulo 30?
4. Let p be an odd prime number, g a primitive root modulo p and ωp a primitive (p− 1)−th root of unity.
(a) (2 points) Define the function χ1 : Z→ C.
χ1(m) = ω
k
p ,
for integers m ≡ gk (mod p) and χ1(m) = 0 if p|m. Justify that χ1 is a Dirichlet Character modulo
p.
1
(b) (2 points) Let χ be some other Dirichlet Character modulo p. Prove there exists an integer t such
that
χ = χt1.
Hint: What happens at the integers congruent to g?
(c) (1 point) Justify that there are exactly p− 1 = ϕ(p) different Dirichlet Characters modulo p. Call
them χ0, χ1, ..., χp−2, with χ0 the principal character modulo p.
(d) (2 points) Let a be an integer that is not a multiple of p. Prove that
χ0(a) + χ1(a) + ...+ χp−2(a) = 0.
Also, determine the value of this sum for the integers that are multiples of p.
(e) (2 points) Let χ be an character that is not the principal character. Prove that
χ(0) + χ(1) + ...+ χ(p− 1) = 0.
Also, find the value of this sum for the principal character χ0.
(f) (1 point) Prove that there exist an infinite number of primes q with q ≡ 1 (mod p).
Hint: Repeat what we did in class for the case 5k+1. The knowledge of the properties of the table
is the content of problem 4.
Remark: This is a particular case of the properties we mentioned in class that the rows and columns
of the table of characters have to satisfy.