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
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.
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