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Math 315; Homework # 5
Due March 4, 2015
1. (Exercise 18.1) Decode the following message, which was sent using the modulus m = 7081 and the exponent k = 1789. (Note that you will rst need to factor
m.)
5192, 2604, 4222
2. (Exercise 20.3) A number a is cal
3
Divisibility
3.1 The definition of divisibility
If m and n are two integers, we say that m divides n (and write m|n)
if n = mk for some integer k. We have already used the terminology
and the notation in some of the discussion on interesting sequences.
4
Primes and Composites
4.1 What is a prime?
A prime p is an integer > 1 whose only positive divisors are 1 and
itself p. A natural number that is not prime is called composite.
The following equivalent formulation is very useful. In the previous chapter,
2
Some Interesting Sequences
2.1 Fibonacci Numbers
Consider the sequence a1 = 1, a2 = 1, a3 = 2, a4 = 3, where the
general term is given by the recursion
an = an1 + an2
for n 3. This is clearly a subset of the natural numbers. We can
easily write a few li
5
Congruences
5.1 Definition of congruence
Given a positive integer n, we say that two integers a and b are
congruent modulo n, written
a b (mod n)
if n divides a b. Thus, for example, 1 is congruent to 6 modulo
5 and 10 is congruent to 3 modulo 7. If an
3 Test 1
3.1 The Questions
There are two parts to the tat. Part A has 10 quations and is mul-
tiple choice. Circle the letter corresponding to your answer. Part B
has 6 quations For PART B, write out your answers justifying your
steps.
3.1.1 Part A
l. The
MATH 315; HOMEWORK # 1
Due Jan 19, 2015
1. (page 11, Exercise 1.3) The consecutive odd numbers 3,5, and 7 are all primes.
Are there innitely many such prime triplets? That is, are there innitely many
prime numbers p so that p + 2 and p + 4 are also primes
Assignment #1 STA457H1S/2202H1S
due Wednesday February 3, 2016
Instructions: Students in STA457S do problems 1 through 3; those in STA2202S do all 4 problems.
1. Daily Japanese yen/US dollar exchange rates (JPY/USD) from Dec. 1, 1978 to Apr.29, 2005
are g
MATH 315; HOMEWORK # 2
Due February 5, 2016
1. (Exercise 6.4 (c) Find all integer solutions of 155x + 341y + 385z = 1. [Hint:
gcd(341, 385) = 11. Write the equation as 155x + 11(31y + 35z) = 1. First solve
the equation 155x + 11u = 1 and then solve 31y +
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MATH 315; HOMEWORK # 1
Due Jan 22, 2016
1. (similar to Exercise 3.3, page 24) Find a formula for all of the points on the
hyperbola x2 2y 2 = 1 whose coordinates are rational numbers.
2. (similar to Exercise 3.4, page 24) The curve y 2 = x3 + 17 contains
MAT 315 PROBLEMS ON CONGRUENCES
(1) Prove that 7x10 + 2 = y 3 has no solutions in x, y Z.
(2) Compute the greatest common divisor of
(a) 5123 and 3215.
(b) 1235 and 2153.
(c) 2351 and 1532.
(3) Find integers x and y so that
(a) 5123x + 3215y = gcd(5123, 3
MAT 315H1S
TUTORIAL 2: SECTION B
(1) Use the Euclidean algorithm to compute the greatest common divisor of the
following pairs of integers:
(a) 235 and 791
(b) 532 and 197
(c) 27128 and 31415
(2) Verify your answers by using the gcd command in SAGE.
(3) I
MATH 315; HOMEWORK # 3
Due Feb. 4, 2015
1. (Exercise 11.2) (a) If m 3, explain why (m) is always even.
(b) (m) is usually divisible by 4. Describe all of the ms for which (m) is
not divisible by 4.
2. (Exercise 11.5 (a) Find x that solves the simultaneous
MATH 315; HOMEWORK # 4
Due Feb. 25, 2015
n
1. (Exercise 14.2) Let Fn = 22 + 1, so, for example, F1 = 5, F2 = 17, F3 = 257
and F4 = 65537. Fermat thought that all of Fn s might be prime, but Euler showed
in 1732 that F5 factors as 641 6700417, and in 1880
UNIVERSITY OF TORONTO
DEPARTMENT OF MATHEMATICS
MAT315H1S
Introduction to Number Theory
Winter Semester 2015
Instructors:
Lectures:
Textbook:
Dr. Henry Kim, BA 6250, 416-978-4443
MWF9
A Friendly Introduction to Number Theory, 4th ed.
by Joseph H. Silverma
MAT 315H1S
TUTORIAL 1: SECTION B
(1) Compute a table of the Fibonacci numbers an for n a multiple of 4 less than 100.
(2) Let
Prove that
and
Deduce that
1+ 5
1 5
=
, =
.
2
2
1
7+3 5
4 =
2
1
73 5 .
4 =
2
4 4 = 3 5.
(3) For any integer n 1, prove that
(
MAT 315H1S
TUTORIAL 1: SECTION C
(1) Consider the sums of cubes
T (n) = 13 + 23 + + n3 .
Compute the value of T (n) for n 50. What do you notice about the values?
(2) Assuming T (n) is given by a polynomial in n, show that the degree of the polynomial is
MAT 315H1S
TUTORIAL 1: SECTION D
(1) Let us set
S1 (n) = 1 + 2 + + n.
If we assume that S1 (n) is given by a polynomial in n, show that the degree of
this polynomial should be 2.
(2) Prove by induction that
1
S1 (n) = n(n + 1).
2
Deduce that S1 (n) is div