Bertrand’s conjecture, proof of

Bertrand’s conjecture, proof of

This is a version of Erdős’s proof as it appears in Hardy and Wright.

We begin with the following lemma.

Lemma

.

Let

n

be a

positive integer

and

p

be a prime. The highest power of

p

dividing

n

!

is

∑

j

⌊

n

p

j

⌋

, where

⌊

x

⌋

is the floor of

x

.

Proof.

Let

p

k

divide

n

!

with

k

as large as possible. Every

p

th term of the

sequence

1

,

…

,

n

is

divisible

by

p

, so contributes a factor to

p

k

. There are

⌊

n

p

⌋

such factors. Every

p

2

th term contributes an extra factor above that, providing

⌊

n

p

2

⌋

new factors. In general, the

p

j

th terms contribute

⌊

n

p

j

⌋

extra factors to

p

k

. So the highest power of

p

dividing

n

!

is

∑

j

⌊

n

p

j

⌋

.
∎

We now prove the

theorem

.

Bertrand’s conjecture.

Let

n

>

2

be a minimal counterexample to the claim. Thus there is no prime

p

such that

n

<

p

<

2

⁢

n

.

In the sequence of primes

2

,

3

,

5

,

7

,

13

,

23

,

43

,

83

,

163

,

317

,

631

,

1259

,

2503

,

each succeeding term is smaller than the double of its

predecessor

. This shows
that

n

≥

2503

.

The binomial expansion (

http://planetmath.org/BinomialTheorem

) of

(

1

+

1

)

2

⁢

n

has

2

⁢

n

+

1

terms and has largest term

(

2

⁢

n

n

)

. Hence

(

2

⁢

n

n

)

≥

4

n

2

⁢

n

+

1

≥

4

n

(

2

⁢

n

)

2

.

For a prime

p

define

r

⁢

(

p

,

n

)

to be the highest power of

p

dividing

(

2

⁢

n

n

)

. To compute

r

⁢

(

p

,

n

)

, we apply the lemma to

(

2

⁢

n

)

!

and

n

!

. We get that

r

⁢

(

p

,

n

)

=

∑

j

≥

1

⌊

2

⁢

n

p

j

⌋

-

2

⁢

∑

j

≥

1

⌊

n

p

j

⌋

=

∑

j

≥

1

(

⌊

2

⁢

n

p

j

⌋

-

2

⁢

⌊

n

p

j

⌋

)

.

The terms of the sum are all 0 or 1 and vanish when

j

>

⌊

log

⁡

(

2

⁢

n

)

log

⁡

p

⌋

, so

r

⁢

(

p

,

n

)

≤

⌊

log

⁡

(

2

⁢

n

)

log

⁡

p

⌋

, that is,

p

r

⁢

(

p

,

n

)

≤

2

⁢

n

.

Now

(

2

⁢

n

n

)

=

∏

p

p

r

⁢

(

p

,

n

)

. By the

inequality

just proved, primes larger than

2

⁢

n

do not contribute to this

product

, and by

assumption

there are no primes between

n

and

2

⁢

n

. So

(

2

⁢

n

n

)

=

∏

1

≤

p

≤

n

p

⁢

prime

p

r

⁢

(

p

,

n

)

.

For

n

>

p

>

2

⁢

n

3

,

3

2

>

n

p

>

1

and so for

p

>

2

⁢

n

3

>

2

⁢

n

we can apply the previous

formula

for

r

⁢

(

p

,

n

)

and find that it is zero. So for all

n

>

4

, the contribution of the primes larger than

2

⁢

n

3

is zero.

If

p

>

2

⁢

n

, all the terms for higher powers of

p

vanish and

r

⁢

(

p

,

n

)

=

⌊

2

⁢

n

p

⌋

-

2

⁢

⌊

n

p

⌋

.
Since

r

⁢

(

p

,

n

)

is at most 1, an

upper bound

for the
contribution for the primes between

2

⁢

n

and

2

⁢

n

3

is the product of all primes smaller than

2

⁢

n

3

. This product is

exp

⁡

(

ϑ

⁢

(

2

⁢

n

3

)

)

, where

ϑ

⁢

(

n

)

is the

Chebyshev function

ϑ

⁢

(

n

)

=

∑

p

≤

n

p

⁢

prime

log

⁡

p

.

There are at most

2

⁢

n

primes smaller than

2

⁢

n

and by the inequality

p

r

⁢

(

p

,

n

)

≤

2

⁢

n

their product is less than

(

2

⁢

n

)

2

⁢

n

. Combining this information, we get the inequality

4

n

(

2

⁢

n

)

2

≤

(

2

⁢

n

n

)

≤

(

2

⁢

n

)

2

⁢

n

⁢

exp

⁡

(

ϑ

⁢

(

2

⁢

n

3

)

)

.

Taking

logarithms

and applying the upper bound of

n

⁢

log

⁡

4

for

ϑ

⁢

(

n

)

(

http://planetmath.org/UpperBoundOnVarthetan

), we obtain the inequality

n

3

⁢

log

⁡

4

≤

(

2

⁢

n

+

2

)

⁢

log

⁡

(

2

⁢

n

)

, which is false for sufficiently large

n

, say

n

=

2

11

. This shows that

n

<

2

11

.

Since the conditions

n

≥

2503

and

n

<

2

11

are incompatible, there are no

counterexamples

to the claim.
∎

References

1

G.H. Hardy, E.M. Wright,

An

Introduction

to the Theory of Numbers

, Oxford University Press, 1938.

Title

Bertrand’s

conjecture

, proof of

Canonical

name

BertrandsConjectureProofOf

Date of creation

2013-03-22 13:18:55

Last modified on

2013-03-22 13:18:55

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Proof

Classification

msc 11N05

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