Math 511 Home
Old Assignments
- Due Monday, Oct. 30
- Show the set of polynomials with rational coefficients is a
domain
(with the usual rules of polynomial addition and
multiplication).
- Show that multiplication, (a,b)´(c,d) = (ac,bd), is well-defined
for
D2*/~ for any domain D.
- Let f: D ® D2*/~ by f
(d) = (d,1) where D is any domain. Show that
f (a´b) = f (a) ´ f (b).
- Compute the greatest common divisor of
x6 + x5 + x + 1
and
x4 + x3 + x2 + 1
in Q[x].
- Compute the greatest common divisor of
x6 + x5 + x + 1
and
x4 + x3 + x2 + 1
in Z2[x].
- Due
Monday,
Oct. 16
- Compute
.
- For what value of a does
not have an inverse?
- Solve
.
- Find the least-squares fit of ax + b = y for the
data
- Which bytes have even and which have odd parity: 01011011, 00100000,
11111111?
- Given that the following byte has even parity, what is the missing
bit: 001?0110.
- What is the check-digit for the ISBN number 0-441-34664-?
- Is 0-445-20977-1 a valid ISBN number?
- Encode 1101 using the Hamming (7,4) code.
- Decode 0010101 using the Hamming (7,4) code.
- For Monday, Sept. 25:
- The whole class should work together
(during
class Friday) to find a dozen values of x for which
Qn(x) =
x2 - n*629287 is 29-smooth for some
positive n.
- due Monday, Sept. 25
- Find a2 = b2 with a ¹ ±b
(mod 899).
- Find a2 = b2 with a ¹ ±
b (mod 374).
- Factor 44243 given the hint that 622 = 8172
(mod 44243)
- Factor 12968419 given the hint that 8242 =
125022 (mod 12968419).
- Prove that if n=2p, where p is an odd prime, then a2 =
b2 (mod n) implies that a = ±b
(mod n).
- Use Fermat's factoring method to factor 320827
- Prove that Fermat's factoring method will work for any odd composite
number.
- Use Kraitchik's factoring method to factor 3247.
- What primes under 30 might factor
Q(x) = x2 - 42448001 for some value of x?
- What values of x (6510 £ x £ 6520)
make Q(x) = x2 - 42448001 29-smooth?
- due Friday, Sept. 15
- Compute 7322 (mod 1000)
- Solve x2 + 10x + 10 = 0 (mod 11)
- Solve x2 + 10x + 8 = 0 (mod 11)
- Solve x2 + 10x + 9 = 0 (mod 12)
- Prove a2 = b2 if and only if a = b or
a = -b in any field.
- due Monday, Sept. 11
- Prove Euler's generalization of Fermat's little theorem.
- Create your own RSA public and private keys (both an
encryption
function f(x) and a decryption function
f -1(x) ). Your
n should be at least 6 digits.
- Exchange a signed secure message with a friend using your
RSA
functions from problem 2. Remember that for a signed message you
first
use your decryption function and then your friend's encryption
function.
- Decrypt your friend's message.
- You have intercepted the message 183 which was encrypted
using the
public key
f(x) = x1951 (mod 12319). What
was the original message?
- Due
Friday, Sept. 1:
(Problems are worth 5 points)
- Find 23-1 (mod 117)
- Find 30-1 (mod 157)
- Find 42-1 (mod 201)
- Solve 41x + 13 = 18x + 27 (mod 117)
- Solve 34x + 17 = 21 - 8x (mod 201)
- Solve 34x + 17 = 20 - 8x (mod 201)
- Compute 2383 (mod 5)
- Compute 24874 (mod 15)
- Prove that if a = b (mod n) and
c = d (mod n) then
a+c = b+d (mod n).
- Prove that if a = b (mod n) and
c = d (mod n) then
ac = bd (mod n).
- Due Friday, August 25: (Problems are worth 10 points)
- For what values of n (2 £
n £
9) is Zn a field?
- Can you conjecture a rule for which values of n make Zn
a field?
- Z10 is not a field. Which elements of Z10 have
multiplicative inverses?
- Z15 is not a field. Which elements of Z15 have
multiplicative inverses?
- Can you conjecture a rule for which elements of Zn have
multiplicative inverses?
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bennett@math.ksu.edu
©1998 Andrew G. Bennett