MATH 467 FACTORIZATION AND PRIMALITY
TESTING, FALL TERM 2011, PROBLEMS 3
Return by Monday 18th September
Lehmans method
The object of this homework is to use Lehmans method to nd factors. The rst
two problems could be done by hand, but it would be more co
Homework 1
CSE/MATH 467.01
Due: 26 August, 2016
Notation Let d 6= 0, 1 Z be square-free, that is, not divisible by any
square of an integer except 1 = 12 . That is, if d = ab2 with a, b integers,
then b2 = 1. Once we have the properties of prime numbers,
Homework 06
Math/CSE 467
Due: 30 September, 2016
1-2. a) Use the Chinese Remainder Theorem for to prove the following generalization: The system of equations
x a1
x a2
.
x ar
mod m1
mod m2
mod mr
with mi N has a solution in N if and only if ai aj mod gcd(
Homework 4
CSE/MATH 467
Due: 16 September, 2016
1 and 2. For b Z and k1 , k2 Z0 , we have the following rules for exponents:
i) bk1 bk2 = bk1 +k2
ii) (bk1 )k2 = bk1 k2
0) Prove directly from the definition of multiplication in Z/mZ that multiplication in
Homework 3
Math 467
Due: 9 September, 2016
1.
a) Show that if a, b N have remainders in the set cfw_1, 4 after division by
5, then so does their product.
b) Show that there are infinitely many primes which have remainders 2 or
3 when divided by 5.
Hint: I
Homework 2
CSE/MATH 467
Due: 02 September, 2016
1. Preview Problem from HW1. Implement the script which is Bressouds Alogrithm 1.7 (Algorithm 1 in our pseudocode available at http:/www.math.psu.edu/wdb/467/pseudocode.
pdf) and test it on the following pai
Homework 5
CSE/MATH 467
Due: 23 September, 2016
1. (Coding problem from last week) Write fast exponentiation code and find
ten 10-digit probable primes (with respect to base 2).
13
2. a) Determine 1313 %10.
13
b) Determine 1313 %15.
3. Recall that we form
Exam 1 Review Sheet
CSE/MATH 467
In-class Exam Date: 14 October, 2016
Definitions to Know: All, especially the important ones of course.
Algorithms to know well enough to explain or answer questions
about.
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
GCD algorithm (
BEGINNING MATHEMATICA PROGRAMMING FOR CSE/MATH 467
W. DALE BROWNAWELL
1. B EGINNING
A
M ATHEMATICA S ESSION
1.1. Using Mathematicas Graphical Interface. On a workstation with Mathematica installed, you can start a Mathematica Xsession after logging on sim
MATH 467 F
ACTORIZATION AND PRIMALITY
TESTING, F
ALL 2011,
PROBLEMS 7
Return by Monday 24th October
The last question on this homework involves a moderate amount of computing, so
I am giving you two weeks to do it.
1. Suppose that p is a prime number and
MATH 467, THE QUADRATIC SIEVE (QS)
We are given an odd number n which we know to be composite and not a perfect power.
The objective is to nd a noncfw_trivial factor of n. A number m P N is called B cfw_smooth when it has no prime
factor exceeding B .
Alg
MATH 467 FACTORIZATION & PRIMALITY
TESTING, FALL 2011, PROBLEMS 9
Return by Monday 7th November
1. Suppose that m and n are coprime positive integers and that the congruences
x2 a
y2 a
(mod m),
(mod n)
have s and t solutions in x and y respectively. Prove
MATH 467 FACTORIZATION AND PRIMALITY
TESTING, FALL 2011, PROBLEMS 8
Return by Monday 31st October
1. Evaluate the following Legendre symbols.
(i)
40000000003
100000000019
,
100000000057
40000000031
(ii)
L
, (iii)
L
40000000003
100000000091
.
L
2. (i) Prov
BEGINNING PARI PROGRAMMING FOR CSE/MATH 467
W. DALE BROWNAWELL
1. I NSTALLATION
You can follow the instructions in the links for various platforms (including
smartphones(!) given by RosettaCode rosettacode.org/wiki/Category:PARI/GP,
which also gives links