A proof that
e
is transcendental
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
Webpage
http://www.math.ufl.edu/
∼
squash/
13 January, 2012
(at
10:25
)
Abstract:
Uses calculus and divisibility to show that
e
is not algebraic. The file has two proofs, a short one due to
Hilbert and a long one, probably of Hermite.
Notation.
For posint
N
, let
≡
N
means “mod
N
congruent”. Let ::
N
:: mean
≡
N
!
ie., congruence mod
N
factorial.
1
:
Lemma.
For
k
a natnum, integral
J
k
:=
R
∞
0
x
k
e
x
d
x
equals
J
k
=
k
!
.
♦
Proof.
IByParts yields
J
k
=
k
·
J
k

1
. And
J
0
= 1.
2
:
Corollary.
For each natnum
M
and intpoly
f
:
Z
∞
0
f
(
x
)
·
x
M
e
x
d
x
::
M
+1
::
f
(0)
·
M
!
.
♦
Hilbert’s proof that
e
is transcendental
The Setup.
FTSOC, suppose
e
is algebraic of de
gree
D
∈
Z
+
. We thus have an intpoly
h
(
x
) :=
B
D
x
D
+
· · ·
+
B
1
x
+
B
0
,
with
B
D
6
= 0
,
such that
h
(
e
) = 0. And
B
0
6
= 0, since
h
() has mini
mal degree.
Proof of transcendentality.
For a posint exponent
r
to
be chosen later, define
Φ(
x
) := [
x

1][
x

2][
x

3]
· · ·
[
x

D
]
,
and
I
u
‘
:=
Z
u
‘
x
r
Φ(
x
)
r
+1
e
x
d
x .
Thus 0 = 0
·
I
∞
0
=
h
(
e
)
I
∞
0
=
U
(
r
) +
L
(
r
) where
we have split each integral into an Upper part and a
Lower part:
U
(
r
) :=
B
0
I
∞
0
+
X
D
K
=1
B
K
e
K
I
∞
K
,
L
(
r
) :=
X
D
K
=1
B
K
e
K
I
K
0
.
We have that
U
(
r
)
r
!
+
L
(
r
)
r
!
=
0
.
The contradiction will come by showing that
U
(
r
)
r
!
is
always a nonzero integer; then showing that
r
can be
chosen large enough that
L
(
r
)
r
!
is less than 1.
Upperbounding
L
(
r
)
.
Over all
x
in the com
pact interval
[
0
..
D
]
, let
A
be an upperbound for
the absvalue of Φ(
x
) and of
x
·
Φ(
x
).
For each
K
∈
[
1
..
D
]
, then,

I
K
0

is bounded by
A
r
+1
.
Let
B
:=
Max
K
∈
[
1
..
D
]

B
K
 ·
e
K
. It follows that
L
(
r
)
6
D
·
BA
r
+1
.
When divided by
r
!, this quantity goes to zero.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 JURY
 Calculus, Algebra, Prime number, Prof. JLF King, eK IK

Click to edit the document details