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Number theory exercises

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Number theory exercises

MAT , fall 

David Pierce

December , , & , 

http:

//mat.msgsu.edu.tr/~dpierce/Dersler/

Exercise . Assuming b > 1, show σ(a)

a < σ(ab)

ab 6 σ(a) · σ(b) ab . Exercise . Show that

a) an odd number with no more than two prime factors cannot be perfect;

b) if a is perfect, then X

d|a

1 d = 2.

Exercise . Prove the following.

a) φ(2n) = φ(n) if n is odd.

b) φ(2 m ) =

( 2 m−1 , if m > 0, 1, if m = 0.

c) If k > 0 and ` > 0, then φ(2 k · 3 ` ) | 2 k · 3 ` .

d) If φ(a) | a, then a is of the form 1 or 2 k or 2 k · 3 ` , where k > 0

and ` > 0.

(2)

Exercise . Prove the converse of Möbius Inversion Theorem; that is, for arithmetic functions F and G, prove

F (a) = X

d|a

µ(d) · G  a d

 =⇒ G(a) = X

d|a

F (d).

Exercise . a) Show X

d|a

µ(d) · σ  a d



= a.

b) Defining Λ(a) = log p if a = p c , where c > 0, and otherwise Λ(a) = 0, show

log a = X

d|a

Λ(d), Λ(a) = − X

d|a

µ(d) · log d.

Exercise . If 0 < k < p, show that p | p k .

Exercise . a) If a ≡ b (mod m) and a ≡ b (mod n), show a ≡ b (mod lcm(m, n)).

b) If k > 0 and a is odd, show a 2

k

≡ 1 (mod 2 k+2 ).

c) If p is an odd prime, let κ(p ` ) = φ(p ` ). Also let

κ(2 ` ) =

( φ(2 ` ), if 0 6 ` 6 2, φ(2 ` )/2, if ` > 2.

Now define generally

κ(a) = lcm

p|a

κ(p a(p) ).

If gcd(a, m) = 1, show a κ(m) ≡ 1 (mod m).

Exercise . Solve Chinese Remainder Theorem problems.

Referanslar

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