Download Integer Divisibility

V. Adamchik
1
Integer Divisibility
Victor Adamchik
Fall of 2005
Lecture 1 (out of seven)
Plan
1. Basics of divisibility
2. Prime numbers
3. Perfect numbers
Notations
- set of integers
- set of positive integers (also
, also )
- set of nonnegative integers (also
0
)
- logical AND
- logical OR
- exist
(existential quantifier)
! - exist exactly one (unique existential quantifier)
- any (universal quantifier)
x - integer part of x (or the floor function)
Basics of divisibility
In this chapter, we will discuss the divisibility of integers, the set of integers is denoted by
. We
will give a few detailed proofs of some of the basic facts about divisibility. Most of the properties
are quite obvious, but it is still a good idea to know how to prove them.
Definition.
An integer b
0 divides another integer a iff
k
that a
k b.
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21-127: Concepts of Mathematics
We also say that b is a factor (or divisor) of a.
One frequently writes b
Example.
3 12
Exercise.
a to indicate that b divides a.
but
5
12
Let a and b be positive integers
and a
b. How many positive integers not
exceeding a are divisible by b? In other words, find such c
that b
Solution. All numbers divisible by b are in the form b k, where k
c
a and b c
. They are positive and do
not exceed a,
0
Therefore, there are floor( ab ) or
b
a
b
k
a
or
0
k
a
b
such integers
Theorem 1. For all integers a, b, c
(1) 1
a,
1
a and a 0.
(2) Reflexivity: a a.
(3) Transitivity: a b
b
c
a c.
(4) Not-quite antisymmetry: a b
(5) if a b
a c
b a
a n b
a
b
a
b.
m c for any integers n and m
Proof.
(1) and (2) follow immediately
a
(3) Given b
Then c
y
x a and c
b
(4) Given b
Then a
y
xy a
x
a
a 1
a 1
a a
y b.
x
x a and a
and therefore x y
or y
x
a
y
a, so a | c
y b.
xy
0,
1 (there are no zero-divisors in the integers). It follows that either y
1. But x
1 implies a
b, and x
1 implies b
a.
x
1
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(5) Given b
x a and c
Consider n b
m
y a.
c
nb
mc
xan
y am
a xn
ym
a nb
mc
It follows
a nb
mc
Application of Theorem 2.
Do there exist integers x, y, and z such that 6 x
9y
15 z
107?
No, they don't, here is the proof by contradiction.
Since 3, 6 and 9 has a common divisor 3 than 3 must divide its linear combination
3
6x
9y
15 z
3 107 which is wrong.
Question. How many divisors does a positive integer have?
Here is a picture of all divisors of integers in range [1, 500]
20
15
10
5
100
200
300
400
500
Primes
Observation. Every positive integer has at least two divisors: 1 and itself
Definition. Integer p
1 is called a prime if its only positive divisors are 1 and p.
Otherwise it is called a composite.
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The number 1 is a special case which is considered neither prime nor composite
The number 2 is also special, it is the only even prime.
Theorem. There are an infinite number of primes
Proof. (by contradiction)
Assume otherwise, say, p1 , …, pn is a complete list of all primes.
Define
p
p1
p2
…
pn
1.
Since this number p is larger than all the pi , it cannot be prime.
But then, there is some prime that divides p. Since our list is supposedly complete that prime must
be, say, pr .
We have that pr
pr
p
p
pr
pr
p
p1 p2 … pn
pr
1
p1 p2 … pn
But then pr
1.
A contradiction.
QED - end of proof ("quod erat demonstrandum").
How would you find (or generate) primes?
Sieve of Eratosthenes:
(Greek astronomer, 195BC)
Write down the integers from 2 to the highest number n you wish to include in the table. Cross out
all numbers
2 which are divisible by 2. Cross out all numbers
3 which are divisible by 3,
then by 5 and so on. Continue until you have crossed out all numbers divisible by
n .
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Why do we stop at
n ?
Because the next number to cross must be n since we cross all numbers with divisors
n .
Goldbach Conjecture (1742)
Prime numbers satisfy many strange and wonderful properties.
Observation:
6
7
8
9
10
18
3 3
5 2
5 3
7 2
7 3
11 7
What about 117 ? Can you represent it as sum of two primes?
Goldbach made the conjecture that every odd number > 5 is equal to the sum of three primes.
Euler replied that Goldbach's conjecture was equivalent to the statement that every even number >
2 is equal to the sum of two primes.
p1
p2
2n
It is known to be true for for numbers through 6 1016 (checked numerically in 2003)
Mersenne numbers
For some years, people believed that if p is prime, then so is 2 p
22
1, 23
1, 25
1:
1, ...
This is not true for all primes, for example
211
1
2047
23 89
Mersenne Conjecture (15 century, by French monk Marin Mersenne) 2 p
p
1 is prime for
2, 3, 5, 7, 13, 17, 19, 31, 67, 127 , 257 and composite for all other positive integers p < 257.
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It took a few of centuries to show that the conjecture was wrong. Only in 1947 the range up to 258
was checked! It turned out that
a) 267
1 and 2257
1 are not primes
b) Mersenne missed p
Definition: When 2 p
61, 89, 107.
1 is prime it is said to be a Mersenne prime.
The largest known Mersenne prime is (2005):
225,964,951
Theorem. If 2 p
1 is prime, then p is prime.
Proof. By contradiction - we assume that 2 p
number, p
1
1 is prime, but p is not prime. Let p be a composite
r s. Consider the following polynomial
xr
s
1
It can be written as
xr
s
1
xs
1 xs
r 1
xs
r 2
...
xs
1
which is easily proved by expanding the right hand side.
Therefore, if p is composite then x p
Contradiction to our assumption that 2 p
1 is composite, so is 2 p
1, since it's divisible by 2s
1.
1 is prime. QED.
Perfect numbers
Definition. An positive integer is a perfect number if it equals the sum of its proper divisors (not
including itself).
The first few perfect numbers are
6=1+2+3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 = ...
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Question. What is the next perfect number? It seems it should not be a problem to answer this by
writing Java or C program.
Question. Are they all even?
This question is much much harder.... It is not known if any odd perfect numbers exist.
All even perfect number are in the form
6=1+2+3
=2*3
28 = 1 + 2 + 3 + 4 + 5 + 6 +7 = 4 * 7
496 = 1 + 2 + 3 +... + 31
= 16 * 31
8128 = 1 + 2 + 3 +... + 127
= 64 * 127
Generally,
2n
1
2n
1 , when n is prime
This is a relation between the perfect and the Mersenne primes. So the search for Mersenne primes
is also the search for even perfect numbers!