Pries: 405 Number Theory, Spring 2012. Homework 4.
Due: Friday 2/17.
Modular arithmetic and Euler’s Theorem
Read: Stein sections 2.5, start 4.1.
Let p be a prime number. Do 8 of the following problems.
1. Stein 2.20, 2.26, 2.32.
2. Solve x86 ≡ 6 mod 29.
3. Describe all integers m such that φ(m) is not a multiple of 4.
4. Describe all integers m such that φ(m) = m/2.
5. Prove x2 ≡ y 2 mod p if and only if x ≡ ±y mod p.
6. Use the fact that every non-zero element of Z/p is a unit to prove Wilson’s theorem
(p − 1)! ≡ −1 mod p.
7. Let g be a primitive root modulo p. Use the fact that (Z/p)∗ = {g, g 2 , . . . , g p−1 } to
prove Wilson’s theorem (p − 1)! ≡ −1 mod p.
8. If a ∈ Z/p and gcd(i, p − 1) 6= 1, prove that ai is not a primitive root modulo p.
9. Prove that mφ(n) + nφ(m) ≡ 1 mod mn if m and n are relatively prime.
It is not the job of mathematicians to do correct arithmetical operations. It is the job of
bank accountants. Samuil Shchatunovski
1