Number Theory
15 February 2016
Bortrands Postulate: For every nZ>1, there exists a prime p st n<p<2n
Examples:
n
2n
p
2
4
3
3
6
5
4
8
5,7
5
10
7
6
12
7,11
7
14
11,13
8
16
11,13
Legendres Conjecture: There is a prime between every 2 consecutive square
inte
Number Theory
8 February 2016
Definition: an integer is called even if it can be written as 2n, n in Z
Definition: an integer is called odd if it can be written as 2n+1, n in Z
Theorem: Every integer is either odd or even
Proof: Let p Z
If p is even then
2/22/2016
Number Theory
Notes from HW:
tn-1+sn=pn
[(n-1)(n)/2]+n2.=(3n2-n)/2
p0=5,p1,pn
N=6(p1,pn)+5
6(integer)+5-odd
All primes are odd
Every prime has one of these forms
6k, 6k+1, 6k+2, 6k+3, 6k+4, 6k+5
Product of Primes:
(6k+1)(6n+1) 6?+1
(6k+1)(6n+3)
Number Theory
11 February 2016
A) Is the number 32857 prime?
No divisible by 11
B) Is the number 18,989,803,014,663 prime?
No divisible by 3
C) Is the number 4,028,305,786,339 prime?
No divisible by 104,729
nZ
3n one prime 3
3n+1 infinitely many
3n+