Math 328K (Daniel Allcock)
Homework 11, due Friday Apr. 14, 2017
Problem 1. Compute the following exponentials
(a) 848187 mod 1189.
(b) 2340 mod 341. (Caution: the way 340 and 341 appear makes this look like
a straight application of Fermats Little Theore
Math 328K (Daniel Allcock)
Homework 12, due Wednesday April 21, 2017
Problem 1. Factor n = 2821 and use this factorization to show that for any integer
a coprime to n, we have an1 1 mod n. (A brief way to say this is that n is a
Carmichael number.)
Proble
Math 328K (Daniel Allcock)
Homework 10, due Friday Apr. 7, 2017
Problem 1. For every n 1, find all solutions to x2 1 mod 2n . (Hint: for every
n 3, there are four solutions. You will discover that the hypothesis to Hensels
lemma does not hold. Nevertheles
The Chinese Remainder Theorem
We now know how to solve a single linear congruence. In this lecture we
consider how to solve systems of simultaneous linear congruences.
Example. We solve the system 2x 5 (mod 7); 3x 4 (mod 8) of two
linear congruences (in o
Lecture 12
Cyclotomic Polynomials, Primes Congruent to 1 mod n
Cyclotomic Polynomials - just as we have primitive roots mod p, we can have
primitive nth roots of unity in the complex numbers. Recall that there are n
distinct nth roots of unity - ie., solu
Lecture 13
Arithmetic Functions
Today - Arithmetic functions, the Mobius
function
(Definition) Arithmetic Function:
NC
An arithmetic function is a function f :
Eg.
(n) =
d(n) =
(n) =
k (n) =
(n) =
(n) =
the number of primes n
the number of positive diviso
Lecture 22
Four Squares Theorem
Pell-Brahmagupta Equation (continued) - x2 dy 2 = 1,
6= d N,
if (x, y) and
(z, w) solutions then x < z y < w if and only if x + dy < z + dw.
Theorem 78. If (x1 , y1 ) is the least positive solution of x2 dy 2= 1 where 6=d
Lecture 11
Square Roots, Tonelli's Algorithm, Number of Consecutive Pairs of Squares mod p
Defined the Jacobi Symbol - used to compute Legendre Symbol efficiently
(quadratic character)
Eg.
(1729|223) = (168|223) = (4 42|223) = (42|223)
= (2|223)(21|223) =
Lecture 14
Mobius Inversion Formula, Zeta Functions
Recall:
Mobius function (n) and other functions
(
(1)(n)
(n) =
0
if n is squarefree
if n is not squarefree
(n) = number of primes dividing n
U (n) = 1 n
(
1 n=1
1(n) =
0 n>1
U =U =1
(
X
1 n=1
(d) =
0 n>1
Logic at Notre Dame
Logic at Notre Dame
The University of Notre Dames Mathematics and Philosophy Departments
have a long and dedicated involvement with formal logic. In this small
pamphlet, we oer two vignettes from that history, an essay by John
Dawson o
