Homework I Solutions
1.
(
d|m k Z, m = dk
d|n l Z, n = dl
So we get
(
m n = d(k l) d|m n
m + n = d(k + l) d|m + n
2. (a). (99, 20, 101), (143, 24, 145)
(b). (1023, 64, 1025)
2
2 s2 +t2
, 2 ), with
(c). A primitive Pythagorean triple is given by (st, s t
2
HW 4
Problem 1.
(a)
57x 1 mod 71.
Since 71 is prime, there is a unique solution mod 71. Guess x = 1, 2, . . . , 5 and you
get x = 5.
(b) 57 = 3.19, and 81 = 34 . Since the modulus is a prime power, we use our
iterative method to rst solve 57x 1 mod 3. Cle
HOMEWORK 5
DUE: 10/29
For reference, consult Chapter 5 and 6 from J&J. Numbered examples/exercises refer to examples/exercises
from J&J. For problems which partially or fully appear in J&J, or anywhere else, please do not look at solutions; feel free to l
HW 6
When I refer to specic theorem numbers, theyre from the book Elementary Number Theory by Jones and Jones, published by Springer. I sometimes use the shorthand CRT for the Chinese Remainder Theorem. I use the notation Zn for the
integers mod n, and Un
HW 5
When I refer to specic theorem numbers, theyre from the book Elementary Number Theory by Jones and Jones, published by Springer. I sometimes use the shorthand CRT for the Chinese Remainder Theorem. Please let me know if there are
any typos or mistake
HOMEWORK 6
DUE: 11/12
For reference, consult Chapter 6,8 from J&J. Numbered examples/exercises refer to examples/exercises
from J&J. For problems which partially or fully appear in J&J, or anywhere else, please do not look at
solutions; feel free to look
HOMEWORK 3
DUE: 10/3
For reference, consult sections 3.1, 3.2, 4.1 and 4.2 of J&J. Numbered examples/exercises refer to examples/exercises from J&J. For problems which partially or fully appear in J&J, or anywhere else, please do
not look at solutions; fe
LECTURE NOTES 9/19
BENJAMIN BAKKER
The goal of these notes is to generalize our proof of the Fundamental Theorem of Arithmetic to other
rings. Before we understand what that means, consider the following set:
Denition 1. Z[i] = cfw_a + bi|a, b Z C is the
HW 0
Problem 1. We know from the Proposition on page 5 of Hatcher that each choice
of 1 < p gives us a triple of the form
(2pq, p2 q 2 , p2 + q 2 ).
It is also easy to verify that each triple comes from one and only one (p, q). See
Hatcher or try it yours
HOMEWORK 4
DUE: 10/10
For reference, consult Chapters 3 and 4 from J&J. Numbered examples/exercises refer to examples/exercises
from J&J. For problems which partially or fully appear in J&J, or anywhere else, please do not look at solutions; feel free to
HOMEWORK 0
DUE: 9/12
Read the rst 7 pages of Hatchers notes (http:/www.math.cornell.edu/hatcher/TN/TNch0.pdf);
we did this in class. Read further if you like; the rest of these notes generalize to Pythagorean quadruples,
rational points on other quadratic
Homework VII
Due on 11/18/2016
Please make sure you are using the FOURTH EDITION of the textbook,
since exercises on different editions may be different.
1. (a). Use the equality xn 1 = (x 1)(xn1 + xn2 + . + x + 1) to prove
that for odd natural number k,
Homework III
Due on 10/07/2016
Please make sure you are using the FOURTH EDITION of the textbook,
since exercises on different editions may be different.
1. Page 62 Question 8.1
2. Page 62 Question 8.2
3. Page 63 Question 8.5
4. Page 63 Question 8.6
1
Homework II
Due on 09/30/2016 at Recitation
Please make sure you are using the FOURTH EDITION of the textbook,
since exercises on different editions may be different.
1. Page 34, Question 5.1
2. Page 35, Question 5.4
3. Find all integer solutions of the e
Homework IV
Due on 10/14/2016
Please make sure you are using the FOURTH EDITION of the textbook,
since exercises on different editions may be different.
1. Page 63 Question 8.9
2. p is an odd prime number.
(a). Prove there are exactly 2 incongruent soluti
Homework I
Due on 09/23/2016 at Recitation
Please make sure you are using the FOURTH EDITION of the textbook,
since exercises on different editions may be different.
1. Page 19, Question 2.2
2. Page 19, Question 2.6
3. Page 19, Question 2.8
4. Page 24, Qu
Homework V
Due on 10/28/2016
Please make sure you are using the FOURTH EDITION of the textbook,
since exercises on dierent editions may be dierent.
1. Compute the following values of the Eulers Phi Function:
(a).
(125)
(b).
(120)
(c).
(10k ), where k is a
Homework VIII
Due on 12/02/2016
Please make sure you are using the FOURTH EDITION of the textbook,
since exercises on different editions may be different.
1. Page 158 Question 21.1 (a),(b)
2. p is an odd prime. Is p 2 a QR or NR modulo p?
3. Compute the f
Homework VI
Due on 11/04/2016
Please make sure you are using the FOURTH EDITION of the textbook,
since exercises on different editions may be different.
1. Page 88, Question 12.2
2. Page 94, Question 13.3
3. Page 99, Question 14.2
4. Page 99, Question 14.
HW 1
Problem 1. Let S N be a nonempty bounded set and let G be the set of upper
bounds of S. Since S has at least one upper bound, G is non-empty. By the wellordering principle, G has a least element g. We only have to check that g S.
Suppose not; since g
HW 2
Problem 1. The proof is identical to Theorem 2.9 in Jones. Primes can only
be of the form 1 mod 6 or 5 mod 6, since otherwise theyre divisible by 2, 3 or 6.
Suppose there are only a nite number of primes of the form 6x + 5; let them be
p1 , p2 , . .
Quiz I
Math 248
November 19, 2012
Name:
1
1. (a) Show that 3 is a primitive root mod 17. Hint: 27 10 mod 17 and 100 = 2 mod 17.
(b) Find all residues mod 17 of order 4 (you can express your answers as powers of 3).
(c) Find all solutions to x4 1 mod 17 (y
HW 8
When I refer to specic theorem numbers, theyre from the book Elementary Number Theory by Jones and Jones, published by Springer. I sometimes use the shorthand CRT for the Chinese Remainder Theorem. I use the notation Zn for the
integers mod n, and Un
HOMEWORK 7+7 1
2
DUE: 11/26
For reference, consult Chapter 8 from J&J. Numbered examples/exercises refer to examples/exercises
from J&J. For problems which partially or fully appear in J&J, or anywhere else, please do not look at
solutions; feel free to l
Midterm I
Math 248
October 11, 2012
Name:
1
1. (10 points) Let p N be a prime. The order mod p of an integer x Z, x 0 mod p, is
the smallest number d N such that xd 1 mod pit is denoted ordp (x). By Fermats
little theorem, the order exists.
(a) Show that
HOMEWORK 8
DUE: 12/3
For reference, consult Chapter 7 from J&J. Numbered examples/exercises refer to examples/exercises
from J&J. For problems which partially or fully appear in J&J, or anywhere else, please do not look at
solutions; feel free to look at
HW 9
When I refer to specic theorem numbers, theyre from the book Elementary Number Theory by Jones and Jones, published by Springer. I sometimes use the shorthand CRT for the Chinese Remainder Theorem. I use the notation Zn for the
integers mod n, and Un
Final
Math 248
December 19, 2012
Name:
1
1. Recall that Wilsons theorem states, for p a prime, that (p 1)! 1 mod p. Prove
Wilsons theorem using primitive roots.
2
2. (a) Show that x = 10 is a solution to x8 1 mod 73 and x = 2 is a solution to
x9 1 mod 73.
FINAL HOMEWORK
DUE: AT THE FINAL EXAM, 12/19
This nal homework is due in class on the day of the nal, 4:00 Wednesday 12/19 in our usual classroom.
You may consult Jones & Jones chapters 1-10, Davenport, your own notes, past homeworks, solutions, or
my not