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Prime Numbers
from the pre-classical to the avant-garde
Paul Erdős (1913-1996)
The Book
‘You don’t have
to believe in
God, but you
should believe
in The Book’
-Paul Erdős
G.H. Hardy
“There is a very high degree of
unexpectedness, combined with
inevitability and economy.”
A Good Problem (definition)
Easy to state
Hard to solve
Of fundamental importance in mathematics
Euler
2016 = 16 x 126
= 32 x 63
= 32 x 9 x 7
= 25 x 32 x 7
Euclid
325-265 BC, Alexandria
The guy explaining
the maths
‘It’s all Greek to me’
ab + (a-b)2 = (a+b)2
4
4
If a straight line be cut into equal and
unequal segments, the rectangle
contained by the unequal segments of
the whole together with the square on
the straight line between the points of
section is equal to the square on the
half.
Reductio ad absurdum
To prove SOMETHING.
Suppose the OPPOSITE of the
something is true.
Construct a sequence of
implications which lead to a
FALSEHOOD.
Then the OPPOSITE cannot
have been true to start off with.
Hence, SOMETHING is true.
Thomas Hardy: "Reductio ad
absurdum, which Euclid loved
so much, is one of a
mathematician's finest
weapons. It is a finer gambit
than any chess gambit: a chess
player may offer the sacrifice of
a pawn or even a piece, but a
mathematician offers the game.
Euclid’s Proof
Suppose the opposite:
« The primes are not infinite. In fact, I can
make a list of them 2,3,5,7,11,…,P,
where P is the biggest prime. »
Now to find the contradiction…
Paul Erdős, 1913-1996
Child prodigy, he could calculate at
the age of three how many seconds
his family’s friends had lived.
“A mathematician is a machine for
turning coffee into theorems.”
1949 Elementary Proof of the Prime
Number Theorem
Key Idea 1: Square and no square
2016 = 25 × 32 × 7
= 25 × 32 × 50 × 71
= (21 × 30 × 50 × 71) × (24 × 32)
= (21 × 30 × 50 × 71) × (22 × 31)2
SQUARE-FREE PART
All powers are 0 or 1
SQUARE PART
Key Idea 2: Counting divisors
How many multiples of 7 less than 24?
7
14
21
28
35
24
7
= 3.4285714…
24
7
= 3
A stronger Result
1
1
1
1
── + ── + ── + ── + … = ∞
2
3
5
7
?
Not all infinite
sequences add
up to ∞
There are infinitely many primes
Suppose the opposite
«
1
1
1
1
── + ── + ── + ── + … adds up to less than ∞.
2
3
5
7
Then there will be a TAIL
1
1
1
1
1
1
── + ── + … + ── + ── + ── + ── + …
2
3
P
q
r
s
We need to go n primes
along the sequence
for the tail to add to less
than a half
SMALL PRIMES
< 1/2
BIG PRIMES
»
The Proof
Produce statement through logical steps:
X
M
= any number at all
= counts the numbers less than X divisible only
by ‘small primes’
Make the contradiction:
Since X can be anything, pick a value which makes the
statement false
Notation notation notation
p
= biggest of small primes
X
= any number at all
M
= counts the numbers less than X divisible only
by ‘small primes’
k
= any number less than X
In how many ways can numbers less than X be
constructed? Certainly, X cannot be more than this.