Math 104A - Homework 2. Due Wednesday, October 22.
1. Leftovers from the previous week. In case you havent done so last week, turn in:
Problem 4(c) of Section 1.4.
Problem 1 of section 2.1.
2. Linear diophantine equations. Section 2.3, solve problems 1,
Math 104 A - Homework 7. Due Wednesday, December 10.
The date when you have the knowledge to work on each problem is indicated next to it.
1. (Friday, December 5. On the existence of primitive roots and the optimal order of elements.)
Solve Problem 17 (a)
Math 104A - Homework 6. Due Wednesday, December 3.
The date when you have the knowledge to work on each problem is indicated.
1. (Monday, November 24. Primitive roots.) List all primitive roots mod 9 and mod 11,
respectively.
Hint: Once you nd one primiti
Math 104A - Fall 2014 - Midterm II
Name:
Student ID:
Instructions:
Please print your name, student ID.
During the test, you may not use books, calculators or telephones.
Read each question carefully, and show all your work. Answers with no explanation wil
1.
(i) Recall that R comes equipped with the norm
N (a + b 37) = a2 + 37b2
and that the norm is multiplicative:
N (zw) = N (z) N (w).
If 2 were reducible, then we could write
2=
where , are not units in R. Let
= a + b 37
for a, b Z. Taking norms we nd
22
Math 104A - Fall 2014 - Midterm I
Name:
Student ID:
Instructions:
Please print your name, student ID.
During the test, you may not use books, calculators or telephones.
Read each question carefully, and show all your work. Answers with no explanation will
Math 104A - Fall 2014 - Final Exam Solutions
Problem 1.
Solve the quadratic congruence
2x2 5x 4 0
mod 72 .
Solution:We rst solve the congruence
2x2 5x 4 0
mod 7.
We use the quadratic formula
5 1
5 57
=
= x1 = 1, x2 = 6 41 = 6 2 = 5 in Z7 .
x1,2 =
4
4
We u
Math 104A - Homework 4. Due Wednesday, November 12.
1. (Multiplicative functions.) Let k 0, and dene for each positive integer n the function
dk .
k (n) =
d|n,d>0
(i) Show that if (n, m) = 1 then any divisor d > 0 of the product nm can be written as
d = d
Math 104A - Homework 5
Due 7/21
1 Write a code that implements Gaussian quadrature with 3 nodes. The
quadrature is given by
1
f (x)dx a0 f (x0 ) + a1 f (x1 ) + a2 f (x2 ),
1
5
where x0 = 3 , x1 = 0, x2 = 3 and a0 = 9 , a1 = 8 , a2 = 5 . The code
5
5
9
9
s
Math 104A - Homework 3
Due 7/7
2.4.6 Show that the following sequences converge linearly to p = 0. How large
must n be before we have |pn p| 5 102 ?
a pn = 1/n.
Since
|pn+1 0|
1/(n + 1)
n
=
=
1
|pn 0|
1/n
n+1
as n , we have that pn converges linearly to 0
Math 104A - Homework 4
Due 7/14
1 Write a function that takes as input an interval [a, b], a number subintervals
N , and a function f , and outputs the approximate derivative of f at the
nodes xi = a + i h, for i = 0, . . . , N , where h = ba . Use the th
Math 104A - Homework 5. Due Wednesday, November 19.
1. (Quadratic congruences.) In class, we have studied congruences of the type
x2
mod p.
We now discuss arbitrary quadratic congruences.
Let p > 2 be a prime and consider the quadratic polynomial
f (x) =
Solutions
1.
(i) The units in Z36 are obtained by requiring (a, 36) = 1. This yields
U (Z36 ) = cfw_1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35.
(ii) We let x be the inverse of 11. Then
11x 1
mod 36 = 11x = 1 + 36y = 11x + 36(y) = 1.
We run the Euclidean
Math 104A - Homework 1
Section 1.1 - 2, 4a, 4c, 8, 12a, 12b, 24
2 Find intervals containing solutions to the following equations.
(a) f (x) = x 3x = 0.
Since f (0) = 1 and f (1) = 2 , by the intermediate value theorem,
3
there is some solution of f in the
Math 104A: Fall 2012
Homework 1
Due Friday, 10/5/2012 at 5:00 pm
1. For x R, dene li(x) by
x
(1)
li(x) :=
2
1
dt.
log(t)
Prove that
(x)
=1
x li(x)
(2)
lim
if and only if
(x)
= 1,
x x/ log(x)
(3)
lim
where (x) is the number of positive prime numbers x. (
Math 104A: Fall 2012
Homework 5
Due Friday, 11/9/2012 at 5:00 pm
1. Show that if m > 2 and if cfw_a1 , . . . , am Z is a complete residue system for Zm , then
cfw_a2 , . . . , a2 Z is not a complete residue system for Zm .
1
m
2. Prove that if a, b > 0
Math 104A: Fall 2012
Midterm 1 Solutions and Comments
Instructions: Please write your name on your blue book. Make it clear in your blue book
what problem you are working on. Write legibly. This exam is graded out of 100 points.
Following these instructio
Math 104A: Fall 2012
Homework 6
Due Friday, 11/16/2012 at 5:00 pm
1. Let a, b, n Z+ with (a, b) = 1. Prove that there exists x Z such that (ax + b, n) = 1.
(Hint: Consider applying the Chinese Remainder Theorem with moduli given by the set of
primes divid
Math 104A: Fall 2012
Midterm 1
Wednesday, 10/24/2012
Instructions: Please write your name on your blue book. Make it clear in your blue book
what problem you are working on. Write legibly. This exam is graded out of 100 points.
Following these instruction
Math 104A: Fall 2012
Midterm 2
Monday, 11/19/2012
Instructions: Please write your name on your blue book. Make it clear in your blue book
what problem you are working on. Write legibly. This exam is graded out of 100 points.
Following these instructions i
Math 104A: Fall 2012
Midterm 2 Solutions
Monday, 11/19/2012
Instructions: Please write your name on your blue book. Make it clear in your blue book
what problem you are working on. Write legibly. This exam is graded out of 100 points.
Following these inst
Math 104A: Fall 2012
Homework 5
Due Friday, 11/9/2012 at 5:00 pm
1. Show that if m > 2 and if cfw_a1 , . . . , am Z is a complete residue system for Zm , then
cfw_a2 , . . . , a2 Z is not a complete residue system for Zm .
1
m
Solution: Since cfw_a1 , .
Math 104A: Fall 2012
Homework 4
Due Friday, 11/2/2012 at 5:00 pm
1. Find all pairs (x, y ) Z2 such that 19x + 20y = 1909. Which of these pairs satisfy
x, y > 0?
2. Let p > 0 be prime and let a, b Z. Prove that (a + b)p ap + bp (mod p). (Hint:
Binomial The
Math 104A: Fall 2012
Homework 2
Due Friday, 10/12/2012 at 5:00 pm
1. Dene the Fibonacci numbers Fn by F1 = F2 = 1 and Fn = Fn1 + Fn2 for n > 2. Prove
that no two successive Fibonacci numbers Fn and Fn+1 have a common divisor a > 1.
Solution: For n 2, dene
Math 104A: Fall 2012
Homework 1
Due Friday, 10/5/2012 at 5:00 pm
1. For x R, dene li(x) by
x
(1)
li(x) :=
2
1
dt.
log(t)
Prove that
(2)
(x)
=1
x li(x)
lim
if and only if
(3)
(x)
= 1,
x x/ log(x)
lim
where (x) is the number of positive prime numbers x. (
Math 104A: Fall 2012
Homework 2
Due Friday, 10/12/2012 at 5:00 pm
1. Dene the Fibonacci numbers Fn by F1 = F2 = 1 and Fn = Fn1 + Fn2 for n > 2. Prove
that no two successive Fibonacci numbers Fn and Fn+1 have a common divisor a > 1.
2. Prove that
n
m=1
1
m
Math 104A: Fall 2012
Homework 3
Due Friday, 10/19/2012 at 5:00 pm
1. Evaluate (4665, 12705) and express the result as a linear combination of 4665 and 12705
with coecients in Z.
Solution: We apply the Euclidean Algorithm to get that
12705 = 2(4665) + 3375
Math 104A: Fall 2012
Homework 4
Due Friday, 11/2/2012 at 5:00 pm
1. Find all pairs (x, y ) Z2 such that 19x + 20y = 1909. Which of these pairs satisfy
x, y > 0?
Solution: We have that 19(1) + 20(1) = 1, which implies that 19(1909) + 20(1909) =
1909. Since