Introductory Number Theory. Homework #1
Some solutions and grading notes.
1. Textbook, #1.1 (p. 11)
One can make a list of triangular numbers and squares, and see what numbers appear in both, but that might
take a lot of time and actually tell us little.
Introductory Number Theory. Homework #2
All numbers are integers.
1. The Fibonacci sequence cfw_Fn is dened recursively as follows: F1 = 1, F2 = 1, and then Fn = Fn1 + Fn2
if n 3. Prove: gcd(Fn , Fn1 ) = 1for all n N. Use induction!
2. Prime power factor
Introductory Number Theory. Homework #2
Some solutions.
All numbers are integers.
1. The Fibonacci sequence cfw_Fn is dened recursively as follows: F1 = 1, F2 = 1, and then Fn = Fn1 + Fn2
if n 3. Prove: gcd(Fn , Fn1 ) = 1for all n N. Use induction!
2. Pr
Introductory Number Theory. Homework 3
Due: Tuesday, June 7, 2011, 4:45PM
Note: My words to work on your own seem to continue to be ignored. At least, dont make it so obvious! At least,
UNDERSTAND WHAT YOU ARE COPYING! I know you are copying without reall
Introductory Number Theory.
Midterm Exam-Take twoSolutions
Note: Every question either appeared on the study guide or is a particular case of a question appearing there. In
your proofs, there should be no non sequiturs 1 , irrelevancies, etc. Speculative
Introductory Number Theory
MIDTERM EXAM
Solutions
Instructions.
Use your own paper, or the one I provide at the front. Write clearly. Keep dierent exercises separate; that
is, try to nish one exercise before starting another, and make it clear where one
Introductory Number Theory. First week topics and Homework #1
1
On writing mathematics
Part of the objective of this course is to get students to write mathematics as mathematics should be written.
Ideally, on completing this course you should be able to
Introductory Number Theory.Topics for the make up exam
1. Prove: There exists an innity of prime numbers.
2. True or false: n2 81n + 1681 is prime for all n N?
3. Prove that the product of any three consecutive numbers is divisible by 6.
4. Let b N, b 1.
Introductory Number Theory
MIDTERM EXAM
Solutions
Instructions.
Use your own paper, or the one I provide at the front. Write clearly. Keep dierent exercises separate; that
is, try to nish one exercise before starting another, and make it clear where one
Introductory Number Theory.
Midterm Exam-Take twoSolutions
Note: Every question either appeared on the study guide or is a particular case of a question appearing there. In
your proofs, there should be no non sequiturs 1 , irrelevancies, etc. Speculative
Introductory Number Theory. Homework 5
Due: Tuesday, June 21, 2011, 4:45PM
1. Textbook, Exercise 21.3. This exercise has 5 parts.
2. Textbook, Exercise 21.6, parts (b), (c).
3. Let p 3 be prime and let g be a primitive root modulo p. Show there exists k ,
Introductory Number Theory. Homework 3
Due: Tuesday, June 7, 2011, 4:45PM
Note: My words to work on your own seem to continue to be ignored. At least, dont make it so obvious! At least,
UNDERSTAND WHAT YOU ARE COPYING! I know you are copying without reall
Introductory Number Theory. Homework 3
Due: Tuesday, June 7, 2011, 4:45PM
1. Let m, n be integers, not both 0. Consider the set A = cfw_mx + ny : x, y Z. Show that this set coincides
with the set of all multiples of the greatest common divisor of m, n. Th
Introductory Number Theory. Homework #2
Some solutions.
All numbers are integers.
1. The Fibonacci sequence cfw_Fn is dened recursively as follows: F1 = 1, F2 = 1, and then Fn = Fn1 + Fn2
if n 3. Prove: gcd(Fn , Fn1 ) = 1for all n N. Use induction!
2. Pr
Introductory Number Theory. Homework #2
All numbers are integers.
1. The Fibonacci sequence cfw_Fn is dened recursively as follows: F1 = 1, F2 = 1, and then Fn = Fn1 + Fn2
if n 3. Prove: gcd(Fn , Fn1 ) = 1for all n N. Use induction!
2. Prime power factor
Introductory Number Theory. Homework #1
Some solutions and grading notes.
1. Textbook, #1.1 (p. 11)
One can make a list of triangular numbers and squares, and see what numbers appear in both, but that might
take a lot of time and actually tell us little.