Number Theory (MA3Z7) Problem Sheet IVR

Java Python Number Theory (MA3Z7)

Problem Sheet IV

1. Let p be an odd prime and let q = 2/p−1. Use Wilson’s Theorem to prove that

(q!)2 + (−1)q ≡ 0 (mod p).

[Hint: write (p − 1)! = 1 · 2 · · · q(q + 1)· · ·(p − 1) and consider this (mod p).]

2. For arithmetic functions f and g, define the Dirichlet convolution by

Number Theory (MA3Z7) Problem Sheet IVR

Show that if f and g are multiplicative, then so is f ∗ g.

3. Prove that d(n) is odd if and only if n is a square.

4. Prove that

[Hint: the identity   may be useful.]

5. Let f be a polynomial and multiplicative. Show that this forces

f(n) = nk

for some k ∈ N0