Math 42-Number Theory
Problem Set #8
Due Tuesday, April 19, 2011
5. Prove that if p and q are distinct odd primes, then
p1
2
k=1
kq
+
p
q 1
2
=1
p1 q1
p
=
.
q
2
2
Solution: Consider the rectangle with corners (0, 0), (p, 0), (0, q ) and (p, q ). Draw the
Math 42-Number Theory
Problem Set #4
Due Tuesday, March 8, 2011
1. Find all x in Z that satisfy x 14 mod 65 and x 25 mod 93.
2. Find all x in Z that satisfy x 14 mod 65 and x 25 mod 91.
3. Find all x in Z that satisfy 3x 5 mod 23, 5x 7 mod 24 and 7x 3 mod
Math 42-Number Theory
Problem Set #2
Due Tuesday, February 15, 2011
1. Which of the following can you nd in Z7 ? (If you can nd them, give a value or values.)
12
3
4 5, 2, , , 2, 1, 6
25
2. Which of the following can you nd in Z13 ? In Z6 ? (Again, if yo
Math 42-Number Theory
Problem Set #8
Due Tuesday, April 19, 2011
1. Compute
5
k=1
5k
.
11
Use this to determine whether 5 is a square mod 11 or not.
2. Compute
6
k=1
5k
.
13
Use this to determine whether 5 is a square mod 13 or not.
3. Graph the line from
Math 42-Number Theory
Problem of the Day #1
Due Tuesday, February 1, 2011
1. In a game where you can score either a or b points at a time, where a and b are in N and
relatively prime, what is the maximum unattainable score? Give at least two specic exampl
Math 42-Number Theory
Problem Set #1
Due Tuesday, February 8, 2011
1. Use Euclids algorithm to nd the GCD of 29 and 11.
2. Write
29
11
as a simple continued fraction. Compute all the convergents.
3. Find integers x, y such that 29x + 11y = 1.
4. Describe
Math 42-Number Theory
Problem Set #5
Due Thursday, March 17, 2011
1. Find a generator for U29 . Use it to make a table of logarithms. Use your table of logarithms
from problem 1 to solve 13x3 = 21 mod 29.
2. Use your table of logarithms to solve x4 = 7 mo
Math 42-Number Theory
Problem Set #7
Due Thursday, April 7, 2011
1. Factor 30 in Z[i]. Is there an element of Z[i] with norm 30?
2. Factor 65 in Z[i]. Is there an element of Z[i] with norm 65?
3. How many ways can 30 and 65 be written as a sum of two squa
Math 42-Number Theory
Problem Set #6
Due Thursday, March 24, 2011
1. List the squares (quadratic residues) in U19 , in U23 and in U29 . How many squares are there in
Up ?
2. Give a characterization of the squares in terms of a generator. That is, which po
Math 42-Number Theory
Problem Set #7
Due Thursday, April 7, 2011
10. Prove that given natural numbers a and b, there exist integers q , r, such that a = bq + r
2a
b
where = 1 and 0 r 2 . Prove in addition, that = (1) b . Here, x means the
greatest integer
Math 42-Number Theory
Problem Set #6
Due Thursday, March 24, 2011
2. Give a characterization of the squares in terms of a generator. That is, which powers of a
generator are squares?
Solution: Even powers of generators are squares.
3. Prove the following
M ATH 42 P RACTICE M IDTERM 2
Name:
F OR F ULL C REDIT
S HOW A LL W ORK
N O C ALCULATORS
1. There is a marching band getting into certain congurations. When they line up in lines of 7, there is one
person left over. When they line up in lines of 8, there
Math 42-Number Theory
Problem Set #1
Due Tuesday, February 8, 2011
8. In a game where you can score a or b points at a time (a, b N), what is the largest unattainable
score? Explain why this score is unattainable. (You dont have to show that it is the lar
Math 42-Number Theory
Problem Set #2
Due Tuesday, February 15, 2011
For problems 4, 5 and 6, either prove the statement is true or give an example showing that it is
false (a counterexample).
4. If a and b are elements of Um , then a + b is in Um .
Soluti
Math 42-Number Theory
Problem Set #3
Due Thursday, February 24, 2011
2. Prove that if a b mod m and c d mod m, then ac bd mod m.
Solution: If a b mod m and c d mod m, then by denition, m | (b a) and m | (d c).
In other words, there are integers k and such
Math 42-Number Theory
Problem Set #4
Due Tuesday, March 8, 2011
8. Prove that if g is a generator for Up , then g k has order
prime.
p1
d,
where d = (p 1, k ). Here, p is
Solution: Suppose (g k )x 1 mod p. Then in fact g kx 1 mod p, and by something we
pr
Math 42-Number Theory
Problem Set #5
Due Thursday, March 17, 2011
7. Prove that if u1 and u2 are elements of Um with orders n1 and n2 respectively and (n1 , n2 ) = 1,
then the order of u1 u2 is n1 n2 .
Solution: Suppose (u1 u2 )k 1 mod m. Then consider (u
Math 42-Number Theory
Problem Set #3
Due Thursday, February 24, 2011
1. List the squares in Z5 , Z7 , Z11 and Z13 . How many squares are there in each case? Make a
general statement about the number of squares in Zp when p is prime.
2. Prove that if a b m