Wolstenholme’s theorem

Wolstenholme’s theorem

We want to show first that the

harmonic number

H

n

=

:

1

+

1

2

+

1

3

+

…

+

1

n

=

∫

0

1

1

-

x

n

1

-

x

d

x

is never an

integer

(

n

>

1

).

Denote by

p

the greatest

prime number

not exceeding

n

.  By Bertrand’s postulate there is a prime

q

with

p

<

q

<

2

⁢

p

.  Therefore we have

n

<

2

⁢

p

.  If

H

n

were an integer, then the sum

n

!

⁢

H

n

=

∑

i

=

1

n

n

!

i

had to be

divisible

by

p

.  However its addend

n

!

p

is not divisible by

p

but all other addends are, whence the sum cannot be divisible by

p

.  The contradictory situation means that

H

n

is not integer when

n

>

1

.

Theorem (Wolstenholme).

If

p

is a prime number

greater

than 3, then the

numerator

of the harmonic number

H

p

-

1

=

1

+

1

2

+

1

3

+

…

+

1

p

-

1

is always divisible by

p

2

.

Proof.

Consider the

polynomial

f

(

x

)

=

:

(

x

-

1

)

(

x

-

2

)

⋯

(

x

-

p

+

1

)

.

One has

f

⁢

(

0

)

=

(

p

-

1

)

!

=

f

⁢

(

p

)

(1)

and

f

⁢

(

x

)

=

x

p

-

1

+

a

1

⁢

x

p

-

2

+

a

2

⁢

x

p

-

3

+

…

+

a

p

-

2

⁢

x

+

(

p

-

1

)

!

(2)

where

a

1

,

a

2

,

…

,

a

p

-

2

are integers.  Because

1

,

2

,

…

,

p

-

1

form a set of all modulo

p

incongruent roots of the Fermat’s

congruence

(

http://

planetmath

.org/FermatsTheorem

)

x

p

-

1

≡

1

(

mod

p

)

,  one may write the

identical congruence

x

p

-

1

-

1

≡

x

p

-

1

+

a

1

⁢

x

p

-

2

+

a

2

⁢

x

p

-

3

+

…

+

a

p

-

2

⁢

x

+

(

p

-

1

)

!

(

mod

p

)

.

(3)

It may be written by Wilson’s theorem

(

p

-

1

)

!

≡

-

1

(

mod

p

)

as

a

1

⁢

x

p

-

2

+

a

2

⁢

x

p

-

3

+

…

+

a

p

-

2

⁢

x

≡

0

(

mod

p

)

,

(4)

being thus true for any integer

x

.  From (4) one can successively infer that

p

divides all coefficients

a

i

, i.e. that (4) actually is a formal congruence.

For the

derivative

of the polynomial

f

⁢

(

x

)

one has

f

′

⁢

(

x

)

=

(

x

-

2

)

⁢

⋯

⁢

(

x

-

p

+

1

)

+

…

+

(

x

-

1

)

⁢

⋯

⁢

(

x

-

p

+

2

)

and thus

f

′

⁢

(

0

)

=

-

2

⋅

3

⁢

⋯

⁢

(

p

-

1

)

-

…

-

1

⋅

2

⁢

⋯

⁢

(

p

-

2

)

.

(5)

The

Taylor series

(

Taylor polynomial

) of

f

⁢

(

x

)

coincides with

f

⁢

(

x

)

:

f

⁢

(

x

)

=

f

⁢

(

0

)

+

f

′

⁢

(

0

)

1

!

⁢

x

+

f

′′

⁢

(

0

)

2

!

⁢

x

2

+

…

+

f

(

p

-

1

)

⁢

(

0

)

(

p

-

1

)

!

⁢

x

p

-

1

By (1), this equation implies

0

=

f

′

⁢

(

0

)

+

f

′′

⁢

(

0

)

2

⁢

p

+

…

+

f

(

p

-

1

)

⁢

(

0

)

(

p

-

1

)

!

⁢

p

p

-

2

(6)

Since

p

∣

a

p

-

3

=

f

′′

⁢

(

0

)

2

,  one has

p

∣

f

′′

(

0

)

.  It then follows by (6) that

p

2

∣

f

′

(

0

)

.  And since (5) divided by

-

(

p

-

1

)

!

gives

1

+

1

2

+

1

3

+

…

+

1

p

-

1

=

-

f

′

⁢

(

0

)

(

p

-

1

)

!

,

the assertion has been proved.

References

1

L. Kuipers

: “Der Wolstenholmesche Satz”.  –

Elemente der Mathematik

35

(1980).

Title

Wolstenholme’s theorem

Canonical name

WolstenholmesTheorem

Date of creation

2013-03-22 19:14:06

Last modified on

2013-03-22 19:14:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11C08

Classification

msc 11A07

Related topic

HarmonicNumber

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