MATH152 Problem Set 4 Solutions
1. In both of these problems, the basic idea is inverting the Legendre
symbol using quadratic reciprocity.
(a) We use the complete multiplicativity of the Legendre symbol to reduce
to cases n = 1, 2, or an odd prime q . In
MATH 152: MIDTERM
In class: Closed Book and Closed Notes
NOTE: Proofs/explanations are needed for all problems. All the best!
1. Consider the group of reduced residue classes (mod 1001). (Note 1001 = 7 11 13).
What is the largest possible order of an elem
MATH152 Midterm Solutions
1. (a) Suppose p2 |n. Then we can nd an element x (Z/nZ) of order
(p ) = p(p 1). But since p(p 1) does not divide n 1, xn1 1 mod n.
2
(b) (p 1)|(n 1) is necessary because there are elements of order p 1.
Conversely, since the ord
Math 249A Fall 2010:
Transcendental Number Theory
A course by Kannan Soundararajan
A
L TEXed by Ian Petrow
September 19, 2011
Contents
1
Introduction; Transcendence of e and
is algebraic if there exists p Z[x], p = 0 with p() = 0, otherwise is called
tr
Sums of Two Squares
These are notes for my lecture of November 3. I will discuss the sums of two squares
using slightly dierent methods from the book.
Gauss studied binary quadratic forms in his famous book Disquisitiones Arithmeticae . Many of the result
OSTROWSKIS THEOREM
The prime numbers also arise in a very surprising manner, having little to do with
factoring integers. Namely they arise as the possible ways of dening absolute values on
Q. We begin by dening what an absolute value is.
We say that a fu
Sums of Squares (Continued)
Our goal is to give a dierent proof of Corollary 3.2 on page 167 of the book. Along
the way, well make a list of primes in the Gaussian integers. Almost everything would
apply with modication to other positive denite binary qua
Ideals
This is a supplement to the discussion of unique factorization in the book of Niven
and Zuckerman.
A ring is a set R with an addition and multiplication, and special elements 0 and 1.
It is assumed that
a b c a + (b + c)
a+b=b+a
0+a=a+0=a
For every
DIRICHLETS THEOREM ON PRIMES IN PROGRESSIONS, III
K. Soundararajan
In the preceding two articles we obtained Dirichlets theorem for progressions with
common dierence 3, 4, 5, and 8. Now we aim to generalize the ideas behind those proofs
for an arbitrary m
DIRICHLETS THEOREM ON PRIMES IN PROGRESSIONS, IV
K. Soundararajan
In the last article we dene (q ) Dirichlet characters (mod q ), and used these to isolate
the (q ) reduced residues (mod q ). Then the proof of Dirichlets theorem boiled down to
showing tha
MATH 152: MIDTERM SOLUTIONS
K. Soundararajan
NOTE: Proofs/explanations are needed for all problems. All the best!
1. Consider the group of reduced residue classes (mod 1001). (Note 1001 = 7 11 13).
What is the largest possible order of an element of this
MATH 152: PROBLEM SET 3
Due January 31
1. (a) By induction, or otherwise, prove that for all k 3 and odd a we have
k 2
a2
(mod 2k ).
1
(b). By induction, or otherwise, prove that for all k 3 we have
k 3
52
1 + 2 k 1
(mod 2k ).
(c). For k 3 prove that eve
MATH 152: PROBLEM SET 2
Due January 24
1. Prove that 1729 is a Carmichael number.
2. (a). Show that (nm) = n(m) if every prime that divides n also divides m.
(b). If n has k distinct odd prime factors show that 2k |(n).
3. Let m and n be natural numbers a
MATH152 Problem Set 5 Solutions
Reminder: Many problems this week ask to nd a sequence xn so that f (xn )
converges to k in p-adic norm. That is equivalent to showing that the equation
f (x) k mod pn has a solution, call it xn , for all suciently large n.
MATH152 Problem Set 3 Solutions
1. (a) Induction on k .
Case k = 3: clear.
k 3
Inductive step: The inductive hypothesis for k 1 implies a2
= 1+Q2k1 .
Taking squares on both sides completes the proof.
(b) Induction on k again.
Case k = 3: clear.
k 4
Induct
MATH152 Problem Set 2 Solutions
1. First note 1729 = 7 13 19. Since 7 1, 13 1, and 19 1 all divide
1729 1 = 1728, for any b coprime to 1729, b1728 1 modulo 7, 13, and 19
by Fermats little theorem. By the Chinese remainder theorem, we see that
b1728 1 mod
MATH 152: PROBLEM SET 7
Due March 7
1. By considering integrals of rational functions as discussed in class, prove that
1+ 5
2
,
L(1, 5 ) = log
2
5
and that
L(1, 8 ) = .
22
In evaluating the integrals you may nd helpful the substitution y = x + 1/x.
2. An
MATH 152: PROBLEM SET 6
Due February 28
1. Let k be a natural number. Adapt Euclids argument to show that there are
innitely many primes p for which k is a quadratic residue. You may nd it useful
to think of numbers of the form n2 k .
2. Prove using Eucli
MATH 152: PROBLEM SET 5
Due February 14
1. Let x be a non-zero rational number. Prove that
|x|
|x|p = 1.
p
2. In Q consider the open disc centered at 0 and of radius 1; that is, consider
cfw_|x| < 1 for the usual absolute value. The center here is clearly
MATH 152 Problem set 1 solutions
1. Let = a + bi and = c + di. Let q be a Gaussian integer such that
1
N ( q ) 2 . Such q exists because in the complex plane every point (in
particular ) is within 1/ 2 unit distance from a Gaussian integer. Another
way to
MATH 152: PROBLEM SET 1
Due January 17
1. Recall the ring Z[i] of Gaussian integers, which was dened in class. Let N (a +
bi) = a2 + b2 be the norm, also dened in class. Prove that Z[i] has a division
algorithm: that is, given any two Gaussian integers ,
MATH 152: PROBLEM SET 4
Due February 7
1. Let n be a non-zero integer.
(a) Prove that if p1 and p2 are two primes with p1 p2 (mod 4|n|) then
n
p2 .
n
n
(b) If n 1 (mod 4), prove that if p1 p2 (mod |n|) then p1 = p2 .
n
p1
=
2. Let k 2 be a natural number,
DIRICHLETS THEOREM ON PRIMES IN PROGRESSIONS, II
K. Soundararajan
In the previous article we established Dirichlets theorem when the modulus is 4. Let
us now consider a few more cases of that argument until we can see clearly the strategy of
the general p
Solutions of ax2 + by 2 = 1
These notes are for my lecture of October 29, in which I did one of the homework
problems. The rst section is not really needed for the proof in the second section, but
it sheds some light on the matter to know something about
MATH 152 Problem set 1 solutions
1. Factorize n4 + n2 + 1:
n4 + n2 + 1 = n4 + 2n2 + 1 n2
= (n2 + 1)2 n2
= (n2 + n + 1)(n2 n + 1).
Since both factors are greater than 1 when n > 1, it follows that it is not prime.
2. (i) Proof by contradiction: suppose p i
MATH 152: PROBLEM SET 9
Due December 1
1. (a) Let q be a natural number and let c1 , . . . , cq be arbitrary complex numbers.
Prove that
q
q
2
cn (n)
|cn |2 .
= (q )
(mod q ) n=1
n=1
(n,q )=1
(b). Let q be a natural number and for each character (mod q )
MATH 152: PROBLEM SET 8
Due November 17
1. Use partial summation to show that
N
log n = N log N N + 1 +
1
nN
cfw_ t
dt.
t
x
Consider F (x) = 1 cfw_tdt. What can you say about F (x) for large x? Use that
information, and integration by parts to establish t
MATH 152: PROBLEM SET 7
Due November 10
1. Evaluate L(1, 8 ), L(1, 8 ) and L(1, 5 ).
2. In class we dened multiplicative or Dirichlet characters (mod q ). In this
problem you will nd the analogous additive characters (mod q ). These are functions : Z C, n
MATH 152: PROBLEM SET 6
Due November 3
1. Develop the arithmetic of the ring Z[ 2] = cfw_a + b 2 : a, b Z. Show that
there are division and Euclidean algorithms for this ring. What are the irreducibles
and primes in this ring; are they the same? Explain w
MATH 152: PROBLEM SET 5
Due October 27
1. Divide the residue classes 1, 2, . . . , p 1 (mod p) (p an odd prime) into two
nonempty sets S1 and S2 such that the product of two residue classes from the
same set is always in S1 , while the product of an eleme