Paul Erdos - ``Problems worthy of attack prove their worth ...web.math.ucsb.edu/~jcs/Erdos.pdf ·...

Tragic Gifted Inventive Prolific

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

Jordan Schettler

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

Figure: Paul, age 8

Paul Erdos UCSB

Figure: Paul, age 8

Paul Erdos UCSB

Figure: Paul, age 8

Paul Erdos UCSB

Figure: Paul, age 8

Paul Erdos UCSB

Figure: Paul, age 8

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

As a teenager...

p < 4x

for all x > 0.

Paul Erdos UCSB

As a teenager...

p < 4x

for all x > 0.

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

Ramsey’s Theorem

Paul Erdos UCSB

Ramsey’s Theorem

Paul Erdos UCSB

Ramsey’s Theorem

Paul Erdos UCSB

Erdos and company proved

Moreover, 17 is not enough!

Paul Erdos UCSB

Erdos and company proved

Moreover, 17 is not enough!

Paul Erdos UCSB

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

∞∑n=1

µ(n)Mn

E =∞∑

but, in 1948, Erdos proved E is irrational.

Paul Erdos UCSB

∞∑n=1

µ(n)Mn

E =∞∑

Paul Erdos UCSB

∞∑n=1

µ(n)Mn

E =∞∑

Paul Erdos UCSB

∞∑n=1

µ(n)Mn

E =∞∑

Paul Erdos UCSB

∞∑n=1

µ(n)Mn

E =∞∑

Paul Erdos UCSB

∞∑n=1

µ(n)Mn

E =∞∑

Paul Erdos UCSB

Lambert Series

∞∑n=1

f (n)xn

1− xn =∞∑

∑d |n

In particular, x 7→ 1/2 gives

∞∑n=1

µ(n)2−n

1− 2−n =12

and E =∞∑

1− 2−n =∞∑

σ0(n)2n

Paul Erdos UCSB

Lambert Series

∞∑n=1

f (n)xn

1− xn =∞∑

∑d |n

∞∑n=1

µ(n)2−n

1− 2−n =12

and E =∞∑

1− 2−n =∞∑

σ0(n)2n

Paul Erdos UCSB

Lambert Series

∞∑n=1

f (n)xn

1− xn =∞∑

∑d |n

∞∑n=1

µ(n)2−n

1− 2−n =12

and E =∞∑

1− 2−n =∞∑

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

Erdos to English Dictionary

Paul Erdos UCSB

boss |bôs|:

Paul Erdos UCSB

boss |bôs|: a woman.

Paul Erdos UCSB

boss |bôs|: a woman.slave |slav|:

Paul Erdos UCSB

boss |bôs|: a woman.slave |slav|: a man.

Paul Erdos UCSB

Paul Erdos UCSB

Paul Erdos UCSB

Paul Erdos UCSB

Paul Erdos UCSB

Paul Erdos UCSB

Paul Erdos UCSB

Paul Erdos UCSB

Paul Erdos UCSB

Paul Erdos UCSB

Paul Erdos UCSB

to lecture about math.die |di|: to stop doing mathematics.Sam |sam|: the United States.Joe |jo|: the Soviet Union.

Paul Erdos UCSB

die |di|: to stop doing mathematics.Sam |sam|: the United States.Joe |jo|: the Soviet Union.

Paul Erdos UCSB

to stop doing mathematics.Sam |sam|: the United States.Joe |jo|: the Soviet Union.

Paul Erdos UCSB

Sam |sam|: the United States.Joe |jo|: the Soviet Union.

Paul Erdos UCSB

the United States.Joe |jo|: the Soviet Union.

Paul Erdos UCSB

Joe |jo|: the Soviet Union.

Paul Erdos UCSB

the Soviet Union.

Paul Erdos UCSB

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)

Paul Erdos UCSB

Mathematician Erdos Number

Paul Erdos 0Andrew Granville 1

Barry Mazur 2Karl Rubin 3

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

Adebisi Agboola 4

Jordan Schettler ∞

Paul Erdos UCSB

Non-Wiki References

Paul Erdos UCSB

Paul Erdos - ``Problems worthy of attack prove their worth ...web.math.ucsb.edu/~jcs/Erdos.pdf ·...

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