paul erdos - ``problems worthy of attack prove their worth by …jcs/erdos.pdf · 2013-11-20 ·...

Tragic Gifted Inventive Prolific

Paul Erdos“Problems worthy of attack prove their worth by fighting back.”

Jordan Schettler

UCSB

3/8/2013

Paul Erdos UCSB


Outline

Tragic

Gifted

Inventive

Prolific

Paul Erdos UCSB


Tragic

Paul Erdos UCSB


Budapest, Hungary...

Figure: Paul, age 8

I Two sisters die of scarletfever in 1913

I A few days later, brotherPaul was born

I Father was captured byRussians in WWI

I “White Terror” in Hungary:Fascism, Anti-Semitism

I Mother gets banned fromteaching for 26 years

Paul Erdos UCSB


Gifted

Paul Erdos UCSB


As a teenager...

Erdos proved∏p≤x

p < 4x

for all x > 0.

“cleaner” proof of Bertrand’s Postulate:

Chebyshev said it, we’ll say it again:There’s always a prime between N and 2N.

Paul Erdos UCSB


City Park, Early 1930’s

Paul Erdos UCSB


Erdos left Hungary in 1934

I He always remained avery devoted son

I Ultimately, he had nopermanent home & fewpossessions

I He ran off of Benzedrine,coffee, and passion

Figure: Paul and Mom Anna

Paul Erdos UCSB


Inventive

Paul Erdos UCSB


Ramsey’s Theorem

Any party with 6 people has a group of 3 who all know eachother or all don’t know each other.

Color edges blue or red in acomplete graph of 6 vertices.

There is always a triangle ofsome color.

A party of 5 is not enough!

Paul Erdos UCSB


Erdos and company proved

Any party with 18 people has a group of 4 who all know eachother or all don’t know each other.

Moreover, 17 is not enough!

Paul Erdos UCSB


Let Mn = 2n − 1 (the nth Mersenne number)

∞∑n=1

µ(n)Mn

=12

E =∞∑

n=1

1Mn

= ??,

but, in 1948, Erdos proved E is irrational.

Paul Erdos UCSB


Lambert Series

∞∑n=1

f (n)xn

1− xn =∞∑

n=1

∑d |n

f (d)

xn

In particular, x 7→ 1/2 gives

∞∑n=1

µ(n)2−n

1− 2−n =12

and E =∞∑

n=1

2−n

1− 2−n =∞∑

n=1

σ0(n)2n

Paul Erdos UCSB


Proofs from THE BOOK

He was not religious, and jokingly referred to God as the SF(Supreme Fascist), a being who selfishly kept the best proofs ofall mathematical theorems in a transfinite BOOK.

Paul Erdos UCSB


Prolific

Paul Erdos UCSB


Erdos Never Slowed Down

I He wrote over 1500 papers (most in history, but 2nd toEuler in number of pages)

I had 511 collaborators (most in history)I

Mathematician Erdos NumberPaul Erdos 0

Andrew Granville 1Barry Mazur 2Karl Rubin 3

Adebisi Agboola 4Jordan Schettler ∞

Paul Erdos UCSB




I had 511 collaborators (most in history)

I




Paul Erdos UCSB





Mathematician Erdos Number

Paul Erdos 0Andrew Granville 1

Barry Mazur 2Karl Rubin 3


Paul Erdos UCSB








Paul Erdos UCSB






Andrew Granville 1

Barry Mazur 2Karl Rubin 3


Paul Erdos UCSB






Andrew Granville 1Barry Mazur 2

Karl Rubin 3Adebisi Agboola 4Jordan Schettler ∞

Paul Erdos UCSB








Paul Erdos UCSB







Adebisi Agboola 4

Jordan Schettler ∞

Paul Erdos UCSB








Paul Erdos UCSB


Non-Wiki References

Paul Erdos UCSB

paul erdos - ``problems worthy of attack prove their worth by …jcs/erdos.pdf · 2013-11-20 ·...

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