DEPARTMENT OF MATHEMATICS
SYLLABUS
Course # & Name:
Mathematics 115A: Number Theory
Recommended Text(s) & Price:
Prepared by:
Jesus deLoera
Matt Nelsenador
Lecture(s)
Rosens Elementary Number Theory and Its
Applications, 6th Edition; Addison-Wesley;
Price
Airamus Jackson, SID 996055554
MAT 115A, SSI16
Homework 1
1. (1.1.5)
of contradiction, suppose that 3is in fact rational
( 3
For sake
Q). Then it follows
that 3 = pq for some p, q N. Note that p = q 3. Let S = cfw_k 3|k, k 3 N. The S 6=
since p = q 3 S
Airamus Jackson, SID 996055554
MAT 115A, SSI16
Homework 4
1. (4.3.2)
x 1 mod 2 = x = 2t + 1 ( t Z)
So,
2t + 1 1 mod 5
2t 0 mod 5
t 0 mod 5
This implies that t = 5u + 0 ( u Z)
So,
x = 2(5u) + 1 = 10u + 1
So,
10u + 1 0 mod 3
10u 1 mod 3
10u 2 mod 3
u 2 mod
Airamus Jackson, SID 996055554
MAT 115A, SSI16
Homework 2
1. (3.3.1) For all problems, we can use the Euclidean Algorithm
(a)
35 = 15 2 + 5
15 = 5 3 + 0
Since 5 is our last non-zero remainder, then it follows that gcd(15, 35) = 5.
(b)
18 = 12 1 + 6
12 = 6
MAT 115A
HW2
Joshua Sumpter
HW 2 Solutions
1
book problems
2.3.18) Use identity (2.2) with n = 4, and then with n = 2, to multiply (10010011)2 and (11001001)2 .
Solution:
We have the following identity: ab = (22n + 2n )A1 B1 + 2n (A1 A0 )(B0 B1 ) + (2n +
def list_of_remainders(base, list_up_to, module):
a = base
k = list_up_to
list = [a]
for i in range(1,k):
list.append(a*(2*i)%module)
return list
def Modular_Exponentiation(base, exponent, module):
N = exponent
#binary representation of N
N_bin = format(N
# Part a:
# The random odd number between j and k is given by
def random_odd_between(j,k):
# random_odd_between function
import random
temp = 2
while temp % 2 = 0 :
temp = random.randint(j,k)
return temp
def Modular_Exponentiation(base, exponent, modulus)
Homework assignment 5 due Monday July 25th.
Instructions: Problems from the book should be written down and turned in together with Problems
1 and 2. To receive extra credit for your homework, do all the extra credit problems, except maybe for one,
and tu
Homework assignment 3 due Monday July 11th.
Instructions: Problems from the book should be written down and turned in together with Problem
2. To receive extra credit for your homework, do all the extra credit problems, except maybe for one,
and turn them
MAT 115A
HW1
Joshua Sumpter
HW 1 Solutions
1b) Given that fn An , show that lim
n
fn+1
= .
fn
fn+1
An
= 1 and that lim
= 1.
n+1
n A
n fn
Proof. By our conjecture, we know that lim
It follows that
An
fn+1
lim
1 = lim
n fn
n An+1
An
fn+1
= lim
n An+1
fn
1
f
Homework assignment 2 due Tuesday July 5th.
Instructions: Problems from the book should be written down and turned in together with Problems
1-3. To receive extra credit for your homework, do all the extra credit problems, except maybe for one,
and turn t
Homework assignment 4 due Monday July 18th.
Instructions: Problems from the book should be written down and turned in together with Problems
1 and 2. To receive extra credit for your homework, do all the extra credit problems, except maybe for one,
and tu
MAT 115A
HW3
Joshua Sumpter
HW 3 Solutions
1
Book Problems
3.5.12) Let n be a positive integer. Show that the power of a prime p occurring in the prime factorization of n! is
given by [n/p] + [n/p2 ] + [n/p3 ] + . . .
Proof. We consider the product form o
Airamus Jackson, SID 996055554
MAT 115A, SSI16
Homework 5
1. (6.3.1)
(a) for n = 6, (n) = 2
for n = 9, (n) = 6
for n = 10, (n) = 4
(b) for n = 6, RRS = cfw_1, 5
for n = 9, RRS = cfw_1, 2, 4, 5, 7, 8
for n = 10, RRS = cfw_1, 3, 7, 9
2. (6.3.4)
E.T. states