Math 350 - Fall 2002 - Number Theory
Homework #4, due Friday, Oct. 11 , 2002.
1. (Schumer 2.4 #3)
Show that infinitely many odd integers of the form n^2+1 are composite.
2. (Schumer 2.5 # 3)
a) Use Fermat's Little Theorem to
determine 3^96 (mod 97), 8^102 (mod 11), and (-5)^12002 (mod 13).
b) Use the Euler-Fermat Theorem to determine 7^1234 (mod 10),
5^1111 (mod 12), and 3^4000 (mod 20)
3. (Schumer 2.5 #11)
Let p be a prime. Show that p^2 divides [ (p-1)^(p-1) - 1 ][(p-1)!+1]
4.
Solve:
2x^780+x^613 == 1 (mod 3)
x^1082+2x^25 == 3 (mod 5)
Here "==" means "congruent to"
5. (Schumer 2.5 #19)
a) Show that the congruence relation
x^5-5x^3+4x == 0 (mod p) has exactly the solutions 0,1,2,p-2, and p-1
(mod p) for p >= 5, p prime.
b) Find all solutions to
2x^5-20x^3+18x == 0 (mod p) for p>=7, p prime.
6. (Schumer 2.6 #2) Show that x^6+98x^5+35x^4+84x^3+21x^2+133x+1 ==
0 (mod 147) is unsolvable by reasoning (mod 7).
7. (Schumer 2.6 #9)
Determine all solutions to x^3+2x-3 == 0 (mod 100)