Second exam for Math 463
Prof. Bernardo brego
May 1st, 2014.
Time limit: 75 minutes. Problems 15 are worth 20 points each, Problem 6 is worth 10 extra points. All
your answers must be justified. Good luck!
In the following problems all variables are integ
Math 342 Number Theory
Solution Quiz 6 (Section D102)
Problem 1 (15 points total)
Dene the following:
(3 point) Quadratic nonresidue mod m
If (a, m) = 1, then a is a quadratic residue mod m if x2 a (mod m) has a
solution. Otherwise, it is a quadratic non
7
More on gcd, lcm and Primality
Theorem 7.1 Suppose that a and b have prime power factorizations a = pa1 pan
1
n
and b = pb1 pbn . Then
1
n
mincfw_a1 ,b1
gcd(a, b) = (a, b) = p1
mincfw_a
pn n ,bn
and
maxcfw_a1 ,b1
lcm(a, b) = [a, b] = p1
maxcfw_an ,b
6
The Fundamental Theorem of Arithmetic
The Fundamental Theorem
The following theorem of Carl F. Gauss was proved earlier.
Theorem 6.1 Let a, b, c be integers. If a|bc and gcd(a, b) = 1, then a|c.
Theorem 6.2 Let p be a prime, let a1 , . . . , ak be integ
0
Note 0
Generalized sum and product
Family: Sometimes a function is called a family, and its domain is called the index set of
the family. If f is a family with the index set with values in a set F (i.e. f : F ), and
the emphasis is on the set F of value
2
The Fibonacci Sequence
Basics
Fibonacci Sequence: This sequence is dened recursively by the following rules:
f0 = 0
f1 = 1
fn = fn1 + fn2 for n 2.
The rst terms are as follows:
fi
value
f1
1
f2
1
f3
2
f4
3
f5
5
f6
8
f7
13
f8
21
f9
34
f10
55
Theorem 2.1
3
Divisibility and Primality
Divisibility
Divides/Divisor/Multiple: For a, b Z, we say that a divides b (or a is a divisor
of b), and write a|b, if there exits q Z such that b = aq. In this case b is called a
multiple of a. Note that a|0 for all a Z since
1
Introduction
Numbers
Integers: We dene Z = cfw_. . . , 2, 1, 0, 1, 2, . . . , to be the set of integers and dene
Z+ = cfw_n Z | n > 0 to be the set of positive integers.
Well-Ordering Property: Every nonempty subset of Z+ has a least element.
Rationals
MATH 342
Assignment 4
Due Monday June 20 by 4pm in the assignment box near the AQ 4000 area.
Last Name:
First Name:
Student #:
Xie
Jianchao
301252416
Please attach this cover sheet and provide answers on additional sheets in order.
(1) How many incongruen
Selected Solutions from Assignment Problems
June 23, 2016
Question 1.4. Prove that if a and b are positive integers satisfying (a, b) =
[a, b], then a = b.
Proof. Let d = (a, b) = [a, b]. Then, as d is a multiple of a and b, d = ax =
by, for some integers
MATH 342
Assignment 1
Due Friday May 27 by 4pm in the assignment box near the AQ 4000 area.
Last Name:
First Name:
Student #:
Please attach this cover sheet and provide answers on additional sheets in order.
(1) Find the greatest common divisor d of 481 a
MATH 342
Assignment 3
Due Monday June 13 by 4pm in the assignment box near the AQ 4000 area.
Last Name:
First Name:
Student #:
Please attach this cover sheet and provide answers on additional sheets in order.
(1) Show that if n is a positive integer with
Solutions to 18.781 Problem Set 2 - Fall 2008
Due Tuesday, Sep. 23 at 1:00
1. (Niven 1.3.39) Prove that
1
1
1
1
1
1
1 1
+ +
=
+
+ +
.
2 3
2007 2008
1005 1006
2008
You may find it easier to prove a general statement!
This problem hints at a general stateme
Math 463 Elementary Number Theory
Homework 4 Solutions
Prof. Arturo Magidin
1. Find all solutions to the congruence 20x 4 (mod 32) by solving the corresponding diophantine
equation (of the form ax + by = c).
Solution. The congruence 20x 4 (mod 32) corresp
115 Homework 6 Solutions
Due Friday November 12
Question 1 (Rosen 4.2.10) Find all integers
modulo 14 and compute it when it exists.
which have an inverse
Solution The integers with inverses modulo 14 are exactly those that are relatively prime to 14. The
1
Problem set 5 - D102
Problem 1.1 Prove the following statements:
If p and 2p + 1 are odd primes, prove that (4p + 2) = (4p) + 2.
If p and 2p1 are odd primes, and n = 2(2p1) then prove that (n) = (n+2).
For each natural number n, (3n) = 3(n) i 3 | n.
11
More on Eulers totient function
As we know
(n) = |cfw_1 k n | (k, n) = 1| =
1.
1kn
(k,n)=1
Notice that for any set E, |E| denotes the number of elements of E and we have
|E| =
1.
xE
Examples: (1) = (2) = 1, (3) = 2 and in general for any prime p, (p) =
10
Eulers Theorem
Eulers Phi or Eulers totient function: For every positive integer n, we let (n)
denote the number of positive integers 1 k n for which (k, n) = 1. I.e.
(n) = |cfw_1 k n | (k, n) = 1|.
&
&
Example: (12) = 4 since it counts the numbers 1,
Math 342 Number Theory
Solution Quiz 6 (Section D103)
Problem 1 (15 points total)
Dene or explain the following:
(3 point) Quadratic residue mod m
If (a, m) = 1, then a is a quadratic residue mod m if x2 a (mod m) has a
solution.
(3 points) Legendre not
Math 342 Number Theory
Solution Quiz 6 (Section D101)
Problem 1 (15 points total)
Dene the following:
(3 point) Quadratic Residue mod a prime p
We say that an integer a is a quadratic residue modulo p if there exists an
integer x such that x2 a (mod a).
Math 342 Number Theory
Solution Quiz 2 (Section D101)
Problem 1 (10 points total)
(a) (6 points) Dene the following:
Divisor: For two integers a and b, we say that a is a divisor of b if there exists
some m Z such that b = am.
lcm of a sequence of integ
Math 342 Number Theory
Solution Quiz 5 (Section D101)
Problem 1 (10 points total)
Dene the following:
(3 point) Primitive root Modulo m
If a has order (m) modulo m, then we call a a primitive root modulo m.
(3 points) Order of an integer relatively prim
Math 342 Number Theory
Quiz 1 Solution (Section D102)
Problem 1 (10 points total)
(a) (6 points) Dene the following:
Quasi-statement: a sentence which is neither always true nor always false.
Instead, its truth or falsity depends upon some variable.
Seq
Math 342 Number Theory
Solution Quiz 5 (Section D103)
Problem 1 (10 points total)
Dene the following:
For a positive integer m; (m)
(m) =| cfw_k N : 1 k m, (k, m) = 1 |
Primitive roots modulo m
x is a primitive root modulo m if ordm a = (m).
Index of a
Math 342 Number Theory
Solution Quiz 1 (Section D101)
Problem 1 (10 points total)
(a) (6 points) Dene the following:
Statement: A statement is a declarative sentence that is either true or false.
Family: A family is a function F where is an index set an
Math 342 Number Theory
Solution Quiz 4 (Section D101)
Problem 1 (10 points total)
Dene the following:
(3 point) Eulers function
If n is a positive integer, (n) is dened to be the number of integers between
1 and n which are coprime to n.
(3 points) Pyth
Math 342 Number Theory
Final Exam - Sample Practice
Summer 2014
Azhvan Ahmady
Name (print):
Signature:
Email:
Final Exam
Problem Score Value
1
7
2
8
3
7
4
7
5
7
6
7
7
7
Total:
50
Re-take Midterm Exam
Problem Score Value
8
5
9
5
10
5
Total:
15
2
Problem 1.
15
Arithmetic Functions
A real or complex valued function of one or several variables where all variables varies in N
is called an Arithmetic function. Therefore a single variable real valued arithmetic function
is a function from N to R.
1
Example 15.1 S
13
Primitive Roots II
Theorem 13.1 (Lagrange) Let p be prime, let n > 0 and let f (x) = an xn +
an1 xn1 + . . . + a1 x + a0 be a polynomial for which an is not divisible by p. Then
f (x) has at most n incongruent roots modulo p.
Proof: We proceed by induc
14
Quadratic Residues
Quadratic Residue: If a and m are relatively prime, then we say that a is a
quadratic residue modulo m if the equation x2 a (mod m) has a solution.
Lemma 14.1 If p is an odd prime and (a, p) = 1 then the equation x2 a (mod p)
has eit