UNIVERSITY OF WATERLOO
MIDTERM EXAMINATION
SPRING TERM 2017
Student Name (Print Legibly)
(family name)
(given name)
Signature
Student ID Number
COURSE NUMBER
PMATH 340
COURSE TITLE
Elementary Number Theory
COURSE SECTION
001
DATE OF EXAM
Friday, June 16th
PMath 340 Elementary Number Theory Spring 2017 Course
Notes
J.C. Saunders
6
6.1
Quadratic Residues
Quadratic Congruences
Lets say we want to solve a quadratic congruence. In other words, we wish to solve
ax2 + bx + c 0 (mod n)
where a, b, c, and n are fix
PMATH 340
Assignment 3 Solutions
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.
One possible answer: We first see if gcd(ni , nj ) | (ai aj ) for all i 6=
PMath 340 Elementary Number Theory Spring 2017 Course
Notes
J.C. Saunders
5
Eulers Function
5.1
Units
We have so far learned how to do addition, subtraction, and multiplication in Zn . This leaves open the question
of division, i.e. is there an answer to
PMATH 340
Assignment 4
Due: Friday, July 7
1. How are the systems Z, Zn , Zp where n is composite and p is prime the
same? How are they different?
2. Find two prime values of p such that 7 19 p is a Carmichael number.
3. Find the solutions to 22x3 16x2 +
PM 450 Solutions to Assignment 4
1. Let > 0, and write Ii = (ai , bi ). I claim that Ii0 = (ai , bi + ) covers [0, 1]. Indeed
let x [0, 1]. Pick a rational point y (x , x + ) Q [0, 1]. For some i, y (ai , bi )
by hypothesis. Hence x Ii0 . Since m ([0, 1])
PM 450 Solutions to Assignment 2
1. We have x = r cos and y = r sin . Therefore
x
y
ur = ux
+ uy
= ux cos + uy sin
r
r
uy
ux
x
y
x
y
urr =
(ux cos +uy sin ) =
cos +
sin = (uxx
+ uxy ) cos + (uyx
+ uyy ) sin
r
r
r
r
r
r
r
= uxx cos2 + 2uxy cos sin + uyy
PM 450 Solutions to Assignment 5
1.
(a) Let gn = infcfw_fi : i n. Then 0 gn fn are non-negative measurable functions which are
monotone increasing to lim inf fn = lim gn =: g. By the Monotone Convergence Theorem
applied to cfw_gn n1 ,
Z
Z
Z
Z
lim inf fn =
PM 450
Assignment 5
Due Wednesday March 15.
Z
1.
(a) Prove that if fn 0 are measurable functions on X, then
Z
lim inf fn lim inf
fn .
(b) Suppose
suppose
that
Z thatZfn 0 are measurable functions on X and fn f a.e. Moreover
Z
Z
lim
fn = f < . Prove that f
PM 450 Solutions to Assignment 1
1.
(a) In standard form, the DE is y 0 = (x, y) = xy + 1. Thus we are seeking a fixed point of the
map
Z x
T f (x) =
1 + tf (t) dt for f C[b, b].
0
When b = 1, we compute for f, g C[1, 1]
Z x
Z x
Z x
t|f (t) tg(t)| dt
1
PM 450 Solutions to Assignment 3
1.
|k|
|k|
1 n+1
= 1 for |k| n + 1; and 2 1
(a) Easy: 2 1 2n+2
n + 2 |k| 2n + 1. Thus 2K2n+1 () Kn () = Vn ().
(b) Clearly Vn is even. Since the coefficient of 1 is always 1, we get
By (a), we have
|k|
2n+2
1
2
=
R
2n
PM 450
1. Define Vn () =
Assignment 3
n+1
X
eik +
k=n1
2n+1
X
2n+2k
n+1
Due Wednesday February 15.
eik + eik .
k=n+2
(a) Prove that Vn () = 2K2n+1 () Kn () (where Kn is the Fejer kernel).
(b) Hence prove that Vn is an even summability kernel.
2. Let f ()
PM 450
Assignment 1
Due Wednesday January 18.
1. Consider the DE: y 0 = 1 + xy and y(0) = 0 for x [b, b], where b > 0.
(a) Reduce this to finding the fixed point of a mapping T . Show that when b = 1, the
map T is a contraction mapping.
(b) Prove that the
PM 450
1.
Assignment 6
Due Wednesday March 29.
(a) Using f () = 3 2 from Assignment 2, Q3, evaluate
X
1
(b) Integrate f and use it to evaluate
.
n8
X
1
.
n6
n=1
n=1
2. Prove that cfw_1, 2 cos n : n 1 is an orthonormal basis for L2 (0, ) with the inner pro
PM 450
Assignment 4
Due Friday March 3.
1. Show that if Q [0, 1] is covered by finitely many intervals I1 , . . . , In , then
Pn
i=1 `(Ii )
1.
2. Let cfw_En n0 be the non-measurable sets constructed in class which partition [0, 1).
(a) Show that if F is
PM 450
Assignment 2
Due Wednesday February 1.
1. Convert the Laplacian u = urr + 1r ur + r12 u to Cartesion coordinates.
Hint: Compute ur , urr , u and u in terms of partials w.r.t. x and y.
2. Let f () = for < .
(a) Compute the Fourier series of f , and
Proof of Prime Factorisation
J.C. Saunders
Theorem 1. For all n N, there there exists a finite set of primes p1 , p2 , ., pk and e1 , e2 , ., ek N such that
n = pe11 pe22 .penn .
Moreover, such a representation of n is unique. This representation is calle
PMath 340 Elementary Number Theory Spring 2017 Course
Notes
J.C. Saunders
3
Congruences
3.1
Introduction to Congruences
Suppose that you want to tell what day of the week it will be 1000 days from now. How would you do it? Well,
you could start off number
PMath 340 Elementary Number Theory Spring 2017 Course
Notes
J.C. Saunders
2
2.1
Prime Numbers
Distribution of Primes
Our study of divisibility leads directly into the study of prime numbers. First, a definition:
Definition 4. A positive integer p greater
PMath 340 Elementary Number Theory Spring 2017 Course
Notes
J.C. Saunders
1
Divisibility
Definition 1. For a pair of integers a, b, we say b divides a or a is a multiple of b or a is divisible by b or b | a if
and only if there exists an integer c such th
Assignment 8
Page 1 of 14
Due: June 28, 11:00am
1. (a) Prove part (2) of the Lemma on page 3 of the Week 8 Slides. That
is, show that if and are formulas of a first order language L
and I = (A, ) is an interpretation of L, then I |= ( ) iff
I |= or I |= .
ur-
(2:223163323331 QU\31%RCEV:\5 ~23 27x
Assignment 3
222(32222 12 (a) Show that cfw_19, Q, ~2(P/\Q) IS an unsamsable 3element set, each
3222222,.22 of whose 2-elemont subsets is satisable.
:3 3333/3533 3322: 33/223 23 2:2 2:. 732222372172 222:2w 22:12 2
gym HoMmmaA WM mm
Assignment 4
1. Use the algorithm described on page 9 of the Week 4 Slides to decide
whether mob of the fOHOWing sets of Hum formulas are satisable,
(a) cfw_19, 3(5 /\ Q), (PA hi) ~3 a), (Q /\ R) 3 P), (P .3 12.).
lm '3 33:33 a a i9 3
a:
Assignment 3 Page 1 of 10 Due: May 24, 11:OOam
1.
(a) Show that cfw_P, Q, -u(P/\Q) is an unsatisable 3elcment set, each
of whose 2-clcmcnt subsets is satisablc.
0? sowL 2 3% a?) T m 9403:
wemma :F ,cfw_?,6\,1L[7A62)( is moms?
cfw_MRS cs 3&va Seth as eCF>s
6? (1642A . Molam mo MGMMOLA
Assignment 1 Page 1 of 10 Due: May 10, 11:00am
1. Let P, Q, and R denote propositional variables. Which of the following
are formulas? For those that are, Show how they are built up from the
propositional variables.
(a) C2
(0)
Assignment 1 Page 1 of 10 Due: May 10, 11:00am
1. Let P, Q, and R denote propositional variables. Which of the following
Recall; are formulas? For those that are, show how they are built up from the
) All $4,0th propositional variables.
u c
vabiu are-Romu
Assignment 4 Page 1 of 9 Due: May 31, 11:00am
1. Use the algorithm described on page 9 of the Week 4 Slides to decide
whether each of the following sets of Horn formulas are satisable.
(a) cfw_13, SAC, (PAR) a S),(Q/\R) + PMP a 12>.
W W Maorhrvi:
Stlfi; (