MATH 115: ELEMENTARY NUMBER THEORY HOMEWORKS #3–4

JOHN VOIGHT

Homework #3 (Due July 12): • §4.2: 1(a)–(c), 6, 10

• §4.3: 4(a)–(c), 12, 14, 15, 16; 4.3A: Let f(x) ∈ ℤ[x] be a polynomial with integral coefficients. For m ∈ ℤ_{>1}, let #X(ℤ/mℤ) denote the number of solutions in ℤ/mℤ of the congruence

f(x) ≡ 0 (mod m).

(a) Prove that if m = m_{1}m_{2}, where gcd(m_{1}, m_{2}) = 1, then

#X(ℤ/mℤ) = #X(ℤ/m_{1}ℤ) · #X(ℤ/m_{2}ℤ).

(b) What can you conclude if gcd(m_{1}, m_{2}) > 1?

• §4.4: 1, 2, 10; 4.4A: Let k ∈ ℤ_{>0}.

(a) Show that the product of any k consecutive integers is divisible by k!. [Hint: Use a binomial coefficient.]

(b) Let f(x) ∈ ℤ[x] be a polynomial with integer coefficients, let r ∈ ℤ. Let f^{(k)}(x) denote the kth derivative of f(x). Show that each coefficient of f^{(k)}(x) is divisible by k!. Conclude that for any r ∈ ℤ, f^{(k)}(r)/k! is an integer.

• §6.1: 3, 12, 14, 16, 25, 33

• §6.3: 2, 5, 9

Homework #4 (Due July 19): • §2.1: 1, 17, 28, 29

• §5.1: 1(c), 2(c), 3(b), 4(d), 11, 12

• §6.2: 1, 3, 18