University of Toronto
Faculty of Arts and Sciences
MAT315H1 - S: Introduction to Number Theory
Fall 2023
Homework 2
1 Problems to be submitted
Make sure you follow all the indications as stated in the syllabus.
1. (5 points) The fundamental parallelogram associated to a Gaussian integer z is the parallelogram
with corners 0, z, iz and (1 + i)z with the segments with (1 + i)z as a vertex removed.
(a) (2 points) Let z ̸= 0 be a non unit Gaussian Integer. Prove that the Gaussian Integers in the
fundamental parallelogram associated to z are a complete residue system modulo z.
(b) (1 point) For each one of z = 1 + i, 3, 1 + 2i write the addition and multiplication table modulo z
using as representatives the integers inside the fundamental parallelogram associated to z.
Note: The numbers 1 + i, 3, 1 + 2i are prime Gaussian Integers. You can assume this.
(c) (1 point) By comparing the tables in the previous part, justify that the integers modulo 1 + i and
1 + 2i are exactly the same as the the rational integers modulo 2 and modulo 5.
(d) (1 point) Does there exists a rational prime p such that the the integers modulo p are the same as
the Gaussian Integers modulo 3?
2. (5 points) Let N be a positive integer. A square modulo N is an class A modulo N such that the
congruence
X2 ≡ A (mod N)
has solutions.
(a) (2.5 points) Let p be an odd prime number and b, c integers such that b2 − 4c is a square modulo
p. Prove that the quadratic formula
x =
−b±√b2 − 4c
2
,
can be used to solve the quadratic congruence
x2 + bx+ c ≡ 0 (mod p).
(b) (2.5 points) Suppose that the quadratic congruence
x2 + bx+ c ≡ 0 (mod p).
is solvable. Prove that b2 − 4c is a square modulo p.
Note: Make sure you understand what the division by 2 means in this context!
3. Solve the following congruences modulo 13 or explain why they have no solutions. Your answers should
be given in the residues 0, 1, ...., 10, 11, 12.
1
(a) (2 points) 5x ≡ 7 (mod 13),
(b) (2 points) 3x ≡ −1 (mod 13),
(c) (2 points) x2 − 4x+ 1 ≡ 0 (mod 13),
(d) (2 points) x2 + 7x ≡ 1 (mod 13),
(e) (2 points) x3 − 6x2 − x+ 2 ≡ 0 (mod 13).
4. Let
ω =
−1 +√−3
2
.
(a) (1 point) Prove that 1− ω is a prime number in the Eisenstein Integers.
(b) (3 points) The fundamental parallelogram associated to a Eisenstein integer z is the parallelo-
gram with corners 0, z, ωz and (1 + ω)z with the segments with (1 + ω)z as a vertex removed.
Using the integers in the fundamental parallelogram associated to (1 − ω)2, find the addition and
multiplication tables of the Eisenstein Integers modulo (1− ω)2.
(c) (1 point) Using the previous table, verify that the Eisenstein Integers modulo (1−ω)2 are the same
as the rational integers modulo 9.
(d) (2 points) Let N be an integer. A cubic residue modulo N is an invertible class A modulo N
such that the congruence
X3 ≡ A (mod N)
is solvable. Find all cubic residues modulo 9.
(e) (3 points) Let X,Y, Z be nonzero Eisenstein Integers relatively prime to 1 − ω. Is it possible to
have
X3 + Y 3 + Z3 = 0?
2 Suggested exercises and problems from the book and other
sources.
Do not submit any of these!
From Number theory of George E. Andrews: Section 4.1: 1, 2, 3, 4, 5, 6, 7; Section 4.2: 1, 2; Section
5.1: 1, 2, 3.
From An Illustrated Theory of Numbers: Chapter 4, 1, 2, 3, 4, 5, 7, 8, 13. Chapter 5, 1, 2, 3, 4, 5, 6,
7, 8, 13, 14, 15, 18.
3 Suggested readings and comments
From An Illustrated Theory of Numbers: Page 140 - 144. (We will discuss this but with less detail).
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