MTH 3527 Number Theory
Quiz 1(Practice)
Show all work - Circle Answers
Name
1. (a) List rst ve odd triangular numbers.
(b) List rst ve even triangular numbers.
2. (a) Let Tn be a triangular number. Prove that 8Tn + 1 is a square of an odd integer.
(b) Acc
MATH 3527
Taught by Sean Clark
Lecture 10
February 11, 2015
The Euler totient function and generalizing Fermats Little Theorem
At this point, it is natural to ask ourselves how to generalize the results we obtained last time for prime moduli
to an arbitra
MATH 3527
Taught by Sean Clark
Lecture 06
January 26, 2015
Primes and the Fundamental Theorem of Arithmetic
Recall that a natural number p is prime if it is only divisible by 1 and itself. Our goal is to now prove the
(perhaps now less obvious) Fundamenta
MATH 3527
Taught by Sean Clark
Lecture 04
January 21, 2015
Greatest common divisor
Last time, we proved the following theorem.
Theorem 1. Let a and b be natural numbers1. There exists a unique pair of integers q (the quotient) and 0 r < b
(the remainder)
MATH 3527
Taught by Sean Clark
Lecture 08
February 4, 2015
Last time: congruences
Let us recall the concept of congruence introduced last time:
Definition 1. Let n N and a, b Z. We say a is congruent to b modulo n if the difference a b is divisible by
n;
MATH 3527
Taught by Sean Clark
Lecture 03
January 15, 2015
Euclids classification of Primitive Pythagorean Triples
Last time, we proved that any primitive Pythagorean triple (a, b, c) can be written such that b is even and a, c
are odd, and that moreover,
MATH 3527
Taught by Sean Clark
Lecture 07
January 28, 2015
Congruence
Before we try to prove more sophisticated results, we should take a step back and introduce a more convenient
language to state things in. This is the language of modular arithmetic, wh
MATH 3527
Taught by Sean Clark
Lecture 12
February 18, 2015
Congruence classes of prime numbers
Theorem 1 (Dirichlets Theorem on Primes in Arithmetic Progressions1). Let a, m N such that gcd(a, m) = 1
Then there are infinitely many primes p such that p a
MATH 3527
Taught by Sean Clark
Lecture 13
February 23, 2015
We are almost in a position to describe the practical application of number theory to cryptography through the
RSA public key cryptosystem. This system depends on two things: the difficulty of in
MATH 3527
Taught by Sean Clark
Lecture 14
February 25, 2015
Computing roots in modular arithmetic
Recall that from last time, we demonstrated the following method for computing roots in modular arithmetic.
Lemma 1 (Method for computing roots). Let n, m N
MATH 3527
Taught by Sean Clark
Lecture 16
March 2, 2015
A bit more on the Rabin-Miller Test
Last time, we proved a result on prime numbers which led us to the following test.
Theorem 1 (Rabin-Miller Test for Composite Numbers). Let n be an odd integer and
MATH 3527
Taught by Sean Clark
Lecture 19
March 16, 2015
Legendre Symbols
Recall that the Legendre symbol is defined as follows.
Definition 1. Let a Z and p be an odd prime. The Legendre symbol of a modulo p is
1
if a is a QR
a
= 1 if a is a NR
p
0
if a
MATH 3527
Taught by Sean Clark
Lecture 17
March 4, 2015
Legendre Symbols
Last time, we proved multiplication rules about quadratic (non-)residues, and observed that
QR QR = NR NR = QR,
1 1 = (1) (1) = 1,
QR NR = NR;
1 (1) = 1.
In particular, we see that t
MATH 3527
Taught by Sean Clark
Lecture 15
February 26, 2015
Primality Testing and Carmichael Numbers
We have now seen some concrete reasons to be interested in finding primes, but as we know primes are difficult
to identify. Is there a practical way to de
MATH 3527
Taught by Sean Clark
Lecture 18
March 5, 2015
Legendre Symbols
Last time, we defined the Legendre symbol:
Definition 1. Let a Z and p be an odd prime. The Legendre symbol of a modulo p is
1
if a is a QR
a
= 1 if a is a NR
p
0
if a is divisible
MATH 3527
Taught by Sean Clark
Lecture 20
March 18, 2015
Two Proofs of Quadratic Reciprocity
Lets remind ourselves what our goal was: we originally wanted to be able to answer the question of when a
quadratic equation x2 n a can have solutions. (In fact,
MATH 3527
Taught by Sean Clark
Lecture 22
March 23, 2015
Writing numbers as a sum of two squares
Now that we have some understanding of what numbers are squares in modular arithmetic, some classical
questions become more open to analysis. In particular, r
MATH 3527
Taught by Sean Clark
Lecture 05
January 22, 2015
Solving Diophantine Equations
Last time, we proved the following theorem.
Theorem 1 (Existence of Diophantine solutions). The equation ax + by = c has integer solutions precisely when
mc
nc
gcd(a,
MATH 3527
Taught by Sean Clark
Lecture 02
January 14, 2015
Natural numbers as real numbers and divisibility
Today, we want to study Pythagorean triples in detail. Before we do that, lets make some of the notions from
the first lecture more precise by clea
MTH 3527 Number Theory
Quiz 1
Show all work - Circle Answers
Name
1. List rst ve even triangular numbers.
2. (a) Let a be an odd integer. Prove that a2 = 8Tk + 1, where Tk is a triangular integer.
(b) Find k so that 112 = 8Tk + 1, where Tk is a triangular
MTH 3527 Number Theory Quiz 2
1. Let a, b, c be positive integers. Either prove or nd a counterexample:
(a) If 6|c and 10|c then 60|c
(b) If a|c and b|c and gdc(a, b) = 1 then ab|c.
2. Use Euclidean algorithm to:
(a) Find gcd(13243, 10336) = g.
(b) Find x
MTH 3527 Number Theory
Quiz 3
Name:
1. Find all incongruent solutions to the following linear congruence:
2. Give a proof by induction: 1 + a + a2 + + an =
1
1an+1
.
1a
66x 100 (mod 121) .
3. For each of the following linear congruences, decide how many n
MTH 3527 Number Theory
Quiz 4
Name:
For this quiz, you always have to state precisely what you are using!
1. Find a number a, such that 0 < a < 97 and a 43961 (mod 97).
2. Solve the following equation x333 5(mod 11). Find all non-congruent solutions.
3. S
MTH 3527 Number Theory
Quiz 5
Name:
1. Using Chinese Remainder Theorem nd x such that
x 27(mod 31), x 50(mod 199) and 0 x < (31)(199) = 6169.
State the theorem, clearly indicate what you are doing at each step, and check your solution.
1
2. You are told t
MTH 3527 Number Theory
Quiz 6
Name:
When using any theorem, make sure that you state it precisely!
1. Determine all positive integers n such that (n) = 40.
2. (a) Prove that there are no primes p such that p 10 (mod 18).
(b) Prove that there is exactly on
MTH 3527 Number Theory
Quiz 7
1. Find 335 (mod 561) using method of successive squaring.
2. Find x such that x35 3(mod 89).
1
Name:
3. Find x such that x35 3(mod 65).
4. Find x such that x35 3(mod 169).
2
MTH 3527 Number Theory
Quiz 6 (Practice)
Name:
1. Consider all positive integers (mod7). How many primes does each equivalence class have?
Explain your statement).
2. Prove that there are innitely many primes congruent to 5 modulo 6.
3. For which values o
MTH 3527 Number Theory
Quiz 6
Name:
When using any theorem, make sure that you state it precisely!
1. Determine all positive integers n such that (n) = 40.
Solution (n) = 40 = 23 5.
n = 41, 2 41, 5 11, 2 5 11, 22 3 11, 23 11, 3 52 , 2 3 52 , 22 52 ,
2. (a
MTH 3527 Number Theory
Quiz 9 (Only some solutions)
Name:
When using any of the theorems, propositions, statements or claims which were already
proved, write the complete statement clearly (not just refering to them by name)!
For example: instead of writi
MTH 3527 Number Theory
Quiz 10
Name:
1. Find all integers n 100 so that 5|(n), where (n) is the Euler function.
Solution Step 1. Find all primes p such that 5 | (pk ).
Let p be prime. Then (pk ) = pk1 (p 1).
If 5|(p) then, one possibility is: p = 5 and k