Math 412: Number Theory
Lecture 9: Euler phi-function and sum and number of
divisors
Gexin Yu
[email protected]
College of William and Mary
Gexin Yu [email protected]
Math 412: Number Theory Lecture 9: Euler phi-function and s
Euler Theorem and Euler phi-function
Euler Theorem: If (a, m) = 1, then a
(m)
⌘ 1 (mod m).
Def: (n) is the number of elements in a reduced system of residues
modulo n. (i.e., the number of coprimes to n in {1, 2, . . . , n}. )
If (m, n) = 1, then (mn) = (m) (n).
If m =
Qr
ni
i=1 pi ,
then (m) = m
Gexin Yu [email protected]
Q
p|m (1
1
p ).
Math 412: Number Theory Lecture 9: Euler phi-function and s
Ex: what are the solutions to the equation (n) = 8, where n is a
positive integer?
Gexin Yu [email protected]
Math 412: Number Theory Lecture 9: Euler phi-function and s
Ex: If p is a prime and n is an integer such that p|(n2 + 1), then p = 2 or
p ⌘ 1 (mod 4). Use this to show that there are infinite many primes of
the form 4k + 1.
Gexin Yu [email protected]
Math 412: Number Theory Lecture 9: Euler phi-function and s
Ex: Thm:
P
d|m
(d) = m.
Gexin Yu [email protected]
Math 412: Number Theory Lecture 9: Euler phi-function and s
Gexin Yu [email protected]
Math 412: Number Theory Lecture 9: Euler phi-function and s
Def: the sum of divisors
P of n:
divisors of n: ⌧ (n) = d|n 1
Gexin Yu [email protected]
(n) =
P
d|n
d and the number of
Math 412: Number Theory Lecture 9: Euler phi-function and s
Def: the sum of divisors
P of n:
divisors of n: ⌧ (n) = d|n 1
Thm:
(p a ) =
p a+1 1
p 1
(n) =
P
d|n
d and the number of
and ⌧ (p a ) = a + 1.
Gexin Yu [email protected]
Math 412: Number Theory Lecture 9: Euler phi-function and s
Thm: if (m, n) = 1, then (mn) = (m) (n) and ⌧ (mn) = ⌧ (m)⌧ (n)
Gexin Yu [email protected]
Math 412: Number Theory Lecture 9: Euler phi-function and s
Thm: if (m, n) = 1, then (mn) = (m) (n) and ⌧ (mn) = ⌧ (m)⌧ (n)
Thm: let n = p1a1 p2a2 . . . psas . Then
(n) =
a +1
s
Y
pj j
j=1
pj
1
1
Gexin Yu [email protected]
and ⌧ (n) =
s
Y
(aj + 1)
j=1
Math 412: Number Theory Lecture 9: Euler phi-function and s