MATH 104B
Winter 2012
HOMEWORK 5
DUE 21 FEBRUARY 2012
Determine whether the integer A is a quadratic residue or nonresidue modulo p for the following integers.
1. A = 500, p = 4219.
2. A = 2003, p = 2011.
3. A = 1903, p = 2011.
4. Let p and q be distinct
QUESTION
(i) State the Law of Quadratic Reciprocity.
(ii) Use (i) to evaluate the Legendre symbol
More precisely, show that
!
3
p
(
=
3
p
when p is an odd prime.
1 if p 1 (modulo 12),
1 if p 5 (modulo 12).
ANSWER
(i) The Law of Quadratic Reciprocity st
QUESTION
(i) Define the Mobius function, (n).
(ii) Let G denote the cyclic group of order n. For each positive integer, d,
dividing n set
f (d) = |cfw_g G|order(g) = d|,
the number of elements of order d in G. Use the Mobius Inversion
Formula to show that
QUESTION
Find all the solutions of each of the following congruences, expressing your
answers in terms of congruence classes, [x]:
(i) 7x 29 (modulo 64),
(ii) 3x5 + x3 + 3001 9 (modulo 14).
ANSWER
(i) Since 7 is odd, the residue class [7] represent a unit
QUESTION
(i) Find HCF(1147,851).
(ii) Find HCF(148,1147,851)
(iii) Find all the integral solutions, x and y, to the linear Diophantine equation
1147x + 851y = 111.
ANSWER
(i) We use the Euclidean algorithm
1147 = 1 851 + 296
851 = 2 296 + 259
296 = 1 259
QUESTION
An integer n 1 is called a Carmichael number if n is not a prime and
an1 1 (modulo n) for all integers a such that gcd(a, n) = 1. Throughout
this question let n denote a Carmichael number.
(i) Show that n cannot be a power of 2.
(ii) Let p be an
QUESTION
Let Un denote the group of units modulo n.
(i) Explain the following terms: (a) g Un is a primitive root and (b) g Un
is a quadratic non-residue.
(ii) Suppose that p = 2m + 1 is a prime for some m > 0. Show rhat g Up
is a quadratic non-residue if
QUESTION
n
For each integer, n 0, define hn = 22 + 1.
(i) Evaluate h0 , h)1, h2 , h3 .
(ii) Show that HCF(hn , hn+t ) = 1 for all n 0 and all t 1. (Hint:
Consider hn+t 2.)
(iii) Use (ii) to give a proof that there exist infinitely many prime numbers.
(Hin
QUESTION
(i) Prove that a positive integer n is
composite if and only if it is divisible
by some prime p such that p n.
(ii) Design a test for deciding when n is the product of at most two primes,
including the possibility that n = p2 for some prime p.
(i
QUESTION
Find all the solutions of each of the following congruences, expressing your
answers in terms of congruence classes, [x]:
(i) 7x 29 mod (51),
(ii) 6(x5 + x3 + 3001) 23 mod (48),
(iii) x2 x + 2 0 mod (4).
ANSWER
(i) Since HCF(7,51)=1 there is exac
QUESTION
Let x be a primitive root modulo p where p is an odd prime and 1 x p1.
(i) Explain why the two sets of congruence classes mod (p),
cfw_[1], [x], [x2 ], . . . , [xp2 ] and cfw_[1], [2], . . . , [p 1]
are equal.
(ii) Using (i) or otherwise, show th
QUESTION
Let p be an odd prime. Let x be a positive integer such that the congruence
class [x] is a generator for Up , the group of units modulo p. If m divides p 1
Q
write (m) for the integer
Y
(m) = 1 + xm + x2m + x3m + . . . + xm(p1)/m)1) = 1 + xm + .
QUESTION
(i) Explaining your reasoning carefully, prove that both 311 and 317 are
primes.
(ii) Stating clearly the theoretical results you have used, calculate the Legendre symbol
317
.
311
(Hint: You may assume that, if p is an odd prime, 2 is a square m
QUESTION
Let a, b, c denote positive integers.
(i) Define the least common multiple, LCM(a, b), and the highest common
factor, HCF(a, b), of a and b.
(ii) Prove the formula
LCM (a, b) =
ab
.
HCF (a, b)
(iii) If we define LCM(a, b, c) and HCF(a, b, c) in a
QUESTION
This question has been broken down into
q a series of cases. However, the
objective is to prove that (n) n (n) when n is a composite integer.
Therefore full marks for parts (i)-(vii) may alternatively be obtained by just
giving an alternative pro
QUESTION
Let p be a prime number and suppose that n! = pe s with HCT(p, s) = 1
where n! is the product of the integers 1, 2, . . . , n, as usual.
(i) Show that
" #
"
#
"
#
"
#
n
n
n
n
e=
+ 2 + 3 + . + r + .
p
p
p
p
where [x] denotes the greatest integer l
MATH 104B
Winter 2012
PRACTICE PROBLEMS
DISCLAIMER: The actual exam questions may have nothing to do with the ones below.
1. Find all the integer solutions to the following diophantine equations or show that no such solutions exist.
(a) x2 + 9 = y 4 ;
(b)
MATH 104B
Winter 2012
PRACTICE PROBLEMS
DISCLAIMER: The actual midterm questions may have nothing to do with the ones below.
1. Compute the continued fraction of the following numbers.
1 3
(a)
2
(b) 6
2. Represent as
(a) [3,
5]
r +s d
t
the following con
MATH 104B
Winter 2012
HOMEWORK 8
DUE FRIDAY 16 MARCH 2012 IN CLASS
1. Formulate and prove a version of Corollary 9.24 for negative discriminants D 1 (mod 4).
Hint: by the proof of Proposition 9.22, H is the subgroup of squares.
2. Let p be a prime number
MATH 104B
Winter 2012
HOMEWORK 2
DUE 31 JANUARY 2012
1. Suppose d = a2 is a perfect square. Find all the integer solutions of the Fermat-Pell
equation
x2 dy 2 = 1.
2. The number
1+ 5
=
2
is called the golden ratio. For each 0 y 20 nd the integer x making
MATH 104B
Winter 2012
QUIZ 1 SOLUTIONS
27 January 2012
1. (a) Show that if x5 + y 5 + z 5 = 0, then
2(x + y + z )5 = 5(x + y )(x + z )(y + z ) (x + y + z )2 + x2 + y 2 + z 2
Use this to show that 5 divides one of the numbers x, y, z.
(b) Show that Fermats
MATH 104B
Winter 2012
HOMEWORK 6
DUE 28 FEBRUARY 2012
1. Determine whether 888 is a quadratic residue or nonresidue modulo the prime 1999 using exclusively the
Legendre symbol.
2. Determine whether 888 is a quadratic residue or nonresidue modulo 1999 by f
MATH 104B
Winter 2012
HOMEWORK 3
DUE 7 FEBRUARY 2012
1. Let A = [a0 , a1 , a2 , . . . , ]. For each n 0 set
pn
= [a0 , a1 , . . . , an ]
qn
where we treat a0 , a1 , . . . as variables, rather than as specic numbers. That is, you dont have to worry
+
about
MATH 104B
Winter 2012
HOMEWORK 4
DUE 14 FEBRUARY 2012
1. These are two identities used by Euler.
(a) Prove that
(x2 + ny 2 )(s2 + nt2 ) = (sx nty )2 + n(tx
(b) Generalize the above to nd an identity of the form
sy )2 .
(ax2 + cy 2 )(as2 + ct2 ) = (?)2 + a
MATH 104B
Winter 2012
HOMEWORK 7
DUE 6 MARCH 2012
1. Show that equivalence and proper equivalence of bqfs are equivalence relations.
2. Show that improper equivalence is not an equivalence relation.
3. Show that equivalent forms represent the same numbers
QUESTION
Let p = am + 1 be a prime, where a 2 and m 1 are integers. Prove that
a must be even and m = 2n for some positive integer, n.
ANSWER
If q is odd then we have the following identity between polynomials with
integral coefficients
tq + 1 = (t + 1)(t
QUESTION
(i) Find gcd(16169,22747).
(ii) Find all the integral solutions, x and y, to the linear Diophantine equation
16169x + 22747y = 69.
ANSWER
(i) We use the Euclidean algorithm.
22747
16169
6578
3013
552
253
46
=
=
=
=
=
=
=
1 16169 + 6578
13156 + 30
QUESTION
(i) Give, without proof, a formula for Eulers function, (n), in terms of the
prime power factorisation of n.
(ii) Let m and n be positive integers such that HCF(m, n) = d. Show that
(d)(mn) = (m)(n)d.
(iii) Hence show that
(mn) (m)(n)
with equali