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PMath 351, Spring Term 2017
Homework Assignment 1 - Solutions
Posted on Thursday, May 4; due on Wednesday, May 10
Problem P0.2 Let A be a non-empty countable set. Suppose that C is a non-empty
set a
A Taste of Number Theory
c Frank Zorzitto
October 6, 2016
ii
Preface
At the University of Waterloo we have been teaching a course in elementary number theory for decades. The idea of offering such a c
PMATH 340
Assignment 2 solutions
Congruences
1. (a) (4 marks) Show that the Diophantine equation x2 + y2 + z2 = 8t + 7 has
no solutions for x, y, z,t Z.
Proof. This is the same as showing that the con
PMATH 340
Assignment 3
(Due Wednesday February 15th at noon)
The Sieve of Eratosthenes
1. (10 marks) It follows from Mertens Second Theorem that the inequality
1
log log n + 6
p prime p
pn
holds for
PMATH 340
Assignment 3
(Due Friday February 17th at noon)
The Sieve of Eratosthenes
1. (10 marks) It follows from Mertens Second Theorem that the inequality
1
log log n + 6
p prime p
pn
holds for any
PMath/AMath 331, Fall 2016 - Assignment 3. Solutions.
Posted on Friday, September 30; due on Friday, October 7.
Topics: Cauchy sequences, Contractive sequences, inner products, Euclidean norm on Rn .
PMath/AMath 331, Fall 2016 - Assignment 1. Solutions
Posted on Friday, September 16. Due on Friday, September 23 at noon.
Topics: Definition of limit, Properties of limits, Bounded sequences, Squeeze
PMath/AMath 331, Fall 2016 - Assignment 10.
Handed out on Friday November 18; due on Friday November 25.
Topics: Finite dimensional vector spaces, pointwise convergence of functions, uniform convergen
PMath/AMath 331, Fall 2016 - Assignment 2. Solutions
Posted on Friday, September 23; due on Friday, September 30 at noon.
Topics: Upper and lower bounds, Least upper bound principle, Monotone converge
PMath/AMath 331, Fall 2016 - Assignment 9.
Posted on on Friday November 11; due on Friday November 18.
Topics: Intermediate Value theorem, continuity in normed vector spaces, uniformly continuous func
PMath/AMath 331, Fall 2016 - Assignment 8.
Posted on Friday November 4; due on Friday November 11.
Topics: Lipschitz constant of matrix transformations, Contractions, properties of continuous function
PMath/AMath 331, Fall 2016 - Assignment 4. Solutions.
Posted on Friday October 7; due on Friday, October 14.
Topics: Abstract normed vector spaces, convergence and completeness in Rn .
Practice proble
PMath/AMath 331, Fall 2016 - Assignment 11.
Posted on Friday November 25; due on Friday December 2.
Topics: Pointwise convergence of functions, uniform convergence of functions.
Practice problems.
Pro
PMath/AMath 331, Fall 2016 - Assignment 4.
Posted on Friday October 7; due on Friday, October 14.
Topics: Abstract normed vector spaces, convergence and completeness in Rn .
Practice problems. (Not to
PMath/AMath 331, Fall 2016 - Assignment 1.
Posted on Friday, September 16. Due on Friday, September 23 at noon.
Topics: Definition of limit, Properties of limits, Bounded sequences, Squeeze theorem,
l
PMath/AMath 331, Fall 2016 - Assignment 5.
Posted on Friday October 14; due on Friday October 21.
Topics: Open and closed subsets of a normed vector space.
Practice problems.
(Not to be submitted for
PMath/AMath 331, Fall 2016 - Assignment 2.
Posted on Friday, September 23; due on Friday, September 30 at noon.
Topics: Upper and lower bounds, Least upper bound principle, Monotone convergence, Bolza
PMath/AMath 331, Fall 2016 - Assignment 3.
Posted on Friday, September 30; due on Friday, October 7.
Topics: Cauchy sequences, Contractive sequences, inner products, Euclidean norm on Rn .
Practice pr
PMath/AMath 331, Fall 2016 - Assignment 7.
Posted on Friday October 28. Due Friday November 4.
Topics: Lipschitz functions.
Practice problems.
Problem P1. Find a bounded continuous function on R that
PMATH 340
Assignment 1 solutions
Divisibility and Factorization
1. Let a, b, c, x, y be integers.
(a) (4 marks) Suppose that c | a and c | b. Prove that c | ax by.
Proof. Since a | b and a | c, there
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laakariin kas. Jai ota ajaakos nuo nuorten lempean viinaan huoneen tulitte. Kayn en niin ei sa
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Assignment 6
cfw_,w 1. Which of the following sets of formulas are consistent? Justify your
Q L cfw_jig; answers, using anything; up to and including all the W'eek 6 content.
(a)
FCKC\Q(1 RMKZMVCQ SKA
Assignment 5
J.
( % Gil?) 1. For each of the following derivamons, provide the justications for each
W line of the derivation.
(a) For any formulas 4,9 and 15!, we have cfw_W70 5
()fOlClQl/C MoMmmd Mal/\WLGuci
Assignment 7
1. Let [Z = cfw_f,g,f, 8,041 wnere f is a unary function symbol, 9 is a
binary function symbol, R is a binary relation symbol? S is a ternary
relation sym
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Assignment 9
1. Translate the following statements about the natural numbers into for
mulas in the language of arithmetic, m. You may use any formulas
or abbreviated formulas
QonJv QM LMVQX 33 n
Assignment 8
1. (a) Prove part (2) of the Lemma on page 3 of the Week 8 Slides. That
~ . is, Show that if (,0 and 1,0 are formulas of a rst order language L:
C 5&7; and I = (A
PMATH 340
Assignment 1 Solutions
Due: Friday, May 26
1. Describe the Remainder Theorem in your own words. For example, when
you take two numbers a and b in the Theorem, you get two other numbers q
and
PMATH 340
Assignment 3
Due: Friday, June 23
1. Describe the method of solving a system of congruences
x a1 (mod n1 )
x a (mod n )
2
2
.
x ak (mod nk ).
in your own words.
2. Describe the method of sol
PMATH 340
Assignment 2
Due: Friday, June 9
1. We have seen one primality test so far. Do you think this specific test is
practical? Why or why not?
2. Describe in your own words what congruence means.
PMATH 340
Assignment 1
Due: Friday, May 26
Hand in your solutions in class on May 26.
For the first three questions, there is more than one correct answer and you dont
need to be mathematically rigoro