HOMEWORK 10 SOLUTIONS
MATH 4713
Question 1 (What do points at innity look like?). Let S 2 = cfw_(x, y, z ) R3 :
x2 + y 2 + z 2 = 1 denote the usual unit sphere in 3-space. Let
H = cfw_(x, y, z ) S 2 : z > 0 and N = (0, 0, 1)
denote the northern hemisphere
HOMEWORK 11 SOLUTIONS
MATH 4713
Question 1. Recall from lecture that the torsion subgroup E (Q)tors of an elliptic
curve E consists of all points P E (Q) such that nP = O for some n N. As we
discussed, we can eectively determine all rational torsion point
HOMEWORK 2
MATH 4713
Question 1.
(1) Prove that gcd(6n + 5, 7n + 6) = 1 for all n Z.
(2) Compute 5500 mod 7.
(3) Prove that 229|132k + 172k for k 1 an odd integer. What about when
k is even? (Note: 229 is prime.)
Question 2. Solve 108x 171 mod 529 for x Z
HOMEWORK 7
MATH 4713
Question 1. Let z = x + yi be a complex number (x, y R). Dene the complex
conjugate to be z = x yi, and the absolute value to be |z| = z z = x2 + y 2 .
(1) Prove that
z+z
x=
2
and
zz
y=
.
2i
Notation: x is the real part of z, denoted
HOMEWORK 9 SOLUTIONS
MATH 4713
Question 1. Recall that 22/7 is a partial convergent to . How close an approximation is it, especially for such a small denominator as 7? Prove that if < a/b < 22/7,
then we must have a denominator b > 106.
Proof. Recall (se
HOMEWORK 7 SOLUTIONS
MATH 4713
Question 1. Let z = x + yi be a complex number (x, y R). Dene the complex
conjugate to be z = x yi, and the absolute value to be |z | = z z = x2 + y 2 .
(1) Prove that
z+z
x=
2
and
zz
y=
.
2i
Notation: x is the real part of
HOMEWORK 4
MATH 4713
Question 1. Recall Wilsons theorem states that (p 1)! 1 mod p for
p a prime. Prove that if n > 4 is composite, then (n 1)! 0 mod n. (If
n = 4, then in fact (n 1)! = 6 2 mod 4, the only exception to this 1 or
0 rule I forgot to mention
HOMEWORK 6 SOLUTIONS
MATH 4713
Question 1. Suppose there is a primitive root g modulo n. We demonstrated in
lecture that there are in fact (n) primitive roots. Suppose h is another primitive
root.1 Prove the change of base formula for the discrete logarit
HOMEWORK 2 SOLUTIONS
MATH 4713
Question 1.
(1) Prove that gcd(6n + 5, 7n + 6) = 1 for all n Z.
Solution. Observe that
6(7n + 6) 7(6n + 5) = 1
This is now in the form of Bezouts identity, and allows us to conclude
that gcd(6n + 5, 7n + 6)|1 because any d t
HOMEWORK 1
MATH 4713
Question 1 (Stein 1.1). Compute gcd(455, 1235) by hand.
Question 2 (Stein 1.2). Use the Sieve of Eratosthenes to make a list of all
primes up to 100.
Question 3 (Stein 1.3). Prove that there are innitely many primes of the
form 6x 1 f
HOMEWORK 1 SOLUTIONS
MATH 4713
Question 1 (Stein 1.1). Compute gcd(455, 1235) by hand.
Question 2 (Stein 1.2). Use the Sieve of Eratosthenes to make a list of all
primes up to 100.
Question 3 (Stein 1.3). Prove that there are innitely many primes of the
f
HOMEWORK 3
MATH 4713
Question 1. Suppose that p is a prime and ap + bp = cp for some a, b, c Z.
Show that p|a + b c.
Question 2. Let p > 2 be a prime. Recall that on Question #6 (2) of HW2
we proved that
(1 + p)p 1 + p2 mod p3 .
Prove that this congruence
HOMEWORK 9
MATH 4713
Question 1. Recall that 22/7 is a partial convergent to . How close an approximation is it, especially for such a small denominator as 7? Prove that if < a/b < 22/7,
then we must have a denominator b > 106.
Question 2. Recall that Eul
HOMEWORK 6
MATH 4713
Question 1. Suppose there is a primitive root g modulo n. We demonstrated in
lecture that there are in fact (n) primitive roots. Suppose h is another primitive
root.1 Prove the change of base formula for the discrete logarithm for a (
HOMEWORK 8
MATH 4713
Question 1. Perform the following evaluations.
(1) Find the continued fraction expansion of 25/41 using the process outlined
in class.
(2) Evaluate [2, 1, 2, 3, 4] as a rational number.
(3) Evaluate [2, 1, 2, 1], where the bar denotes
HOMEWORK 9 SOLUTIONS
MATH 4713
Question 1. Recall that 22/7 is a partial convergent to . How close an approximation is it, especially for such a small denominator as 7? Prove that if < a/b < 22/7,
then we must have a denominator b > 106.
Proof. Recall (se
Math 4713 Assignment 8
Due Monday, Nov. 8, in class, or 2:00 AM Tues. Nov. 9 in
D2L
Problems to Read, think about and read the solutions in the back of the
textbook: Do not turn in your work on these.
Section 4.2: 1, 3, 5, 7, 9, 11, 13, 15.
Section 4.3: 1
Math 4713 Assignment 9
Due Monday, Nov. 22, in class, or 2:00 AM Tues. Nov. 23 in
D2L
Problems to Read, think about and read the solutions in the back of the
textbook: Do not turn in your work on these.
Section 6.1: 9, 11, 15, 19, 25, 47.
Section 6.3: 5,
Math 4713 Assignment 10
Due Friday, Dec. 10, in class, or 2:00 AM Sat. Dec. 11 in
D2L
Problems to Read, think about and read the solutions in the back of the
textbook: Do not turn in your work on these.
Section 7.1: 1, 3, 5, 7, 11, 15, 19, 23.
Section 7.2
Math 4713 Assignment 7
Due Monday, Nov. 1, in class, or 2:00 AM Tues. Nov. 2 in
D2L
Problems to Read, think about and read the solutions in the back of the
textbook: Do not turn in your work on these.
Section 3.7: 1, 3, 5, 11, 13, 15, 19, 21.
Section 4.1:
Math 4713 Assignment 6
Due Friday, Oct. 22, in class, or 2:00 AM Sat. Oct. 23
in D2L
Problems to Read, think about and read the solutions in the back
of the textbook: Do not turn in your work on these.
Section 3.5: 7, 9, 11, 12, 13, 15, 21, 23, 25, 31, 35
Math 4713 Assignment 3
Due Wednesday, Sep. 22, in class, or 2:00 AM
Thurs. Sep. 23 in D2L
Problems to read, think about and read the solutions in the back of the textbook:
Do not turn in your work on these.
Section 2.1: 1, 3, 5, 9, 15, 16, 17.
Also read #
Math 4713 Assignment 4
Due Friday, Oct. 1, in class, or 2:00 AM Saturday, Oct. 2 in
D2L
Problems to read, think about and read the solutions in the back of the textbook:
Do not turn in your work on these.
Section 3.2: 1, 3, 5, 11, 15, 17.
Section 3.3: 1,
Math 4713 Assignment 5
Due Wednesday, Oct. 13, in class, or 2:00 AM
Friday. Oct. 15 in D2L
Problems to Read, think about and read the solutions in the back
of the textbook: Do not turn in your work on these.
Section 3.4: 1, 3, 5, 7, 15, 19.
Written soluti
Math 4713 Assignment 1
Due Monday, Aug. 30, in class, or 2:00 AM Tues. Aug. 31 in
D2L
Odd-numbered problems: Work as many of the early odd numbered problems
(whose complete answers are in the back of the book) in each section covered
as necessary to gain
HOMEWORK 3 SOLUTIONS
MATH 4713
Question 1. Suppose that p is a prime and ap + bp = cp for some a, b, c Z.
Show that p|a + b c.
Proof. Fermats Little Theorem tells us that
ap a mod p,
bp b mod p,
cp c mod p
for any prime p. (Note that this is true for all