aarjaneiro/planetmath huggingface dataset:
fundamental theorem of transcendence

fundamental theorem of transcendence

The tongue-in-cheek given to the fact that if

n

is a nonzero

integer

, then

|

n

|

≥

1

. This trick is used in many

transcendental number

theory proofs. In fact, the hardest step of many problems is showing that a particular integer is not zero.

Title

fundamental theorem of transcendence

Canonical name

FundamentalTheoremOfTranscendence

Date of creation

2013-03-22 13:06:58

Last modified on

2013-03-22 13:06:58

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

5

Author

KimJ (5)

Entry type

Theorem

Classification

msc 11J81

Related topic

EIsTranscendental

Generated on Fri Feb 9 15:35:32 2018 by

LaTeXML
Gelfond-Schneider constant

Gelfond-Schneider constant

The

Gelfond-Schneider constant

2

2

was one of the first numbers to be proven to be transcendental by appying Gelfond’s theorem. However, naming the constant after Gelfond and Schneider comes from Eric Weisstein, with many people preferring to refer to it simply as 2 to the power of

square root of 2

.

Its value in base 10 is approximately 2.66514414269022518865029724987313984827421131371. Its

continued fraction

representation

is neither terminating nor

periodic

, and begins

2

+

1

1

+

1

1

+

1

1

+

1

72

+

⋯

Title

Gelfond-Schneider constant

Canonical name

GelfondSchneiderConstant

Date of creation

2013-03-22 18:54:36

Last modified on

2013-03-22 18:54:36

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11J81

Generated on Fri Feb 9 15:35:34 2018 by

LaTeXML
Gelfond’s constant

Gelfond’s constant

Gelfond’s constant

e

π

was one of the first numbers to be proven to be transcendental by appying Gelfond’s theorem. However, naming the constant after Gelfond comes from Eric Weisstein, with many people preferring to refer to it simply as

e

to the power of

π

.

Its value in base 10 is approximately 23.1406926327792690057290863679485473802661062426. Its

continued fraction

representation

is neither terminating nor

periodic

, and begins

23

+

1

7

+

1

9

+

1

3

+

⋯

Title

Gelfond’s constant

Canonical name

GelfondsConstant

Date of creation

2013-03-22 18:54:34

Last modified on

2013-03-22 18:54:34

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11J81

Generated on Fri Feb 9 15:35:35 2018 by

LaTeXML
Gelfond’s theorem

Gelfond’s theorem

Let

α

and

β

be algebraic over

ℚ

, with

β

irrational

and

α

not equal to 0 or 1. Then

α

β

is transcendental over

ℚ

.

This is perhaps the most useful result in determining whether a number is

algebraic

or transcendental.

The theorem is also known as the

Gelfond-Schneider Theorem

.

Title

Gelfond’s theorem

Canonical name

GelfondsTheorem

Date of creation

2013-03-22 13:24:31

Last modified on

2013-03-22 13:24:31

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 11J81

Synonym

Gelfond-Schneider Theorem

Related topic

LindemannWeierstrassTheorem

Related topic

Irrational

Generated on Fri Feb 9 15:35:38 2018 by

LaTeXML
Liouville number

Liouville number

A

Liouville number

is an

irrational number

l

such that for any

integer

n

>

0

there is a pair of integers

j

and

k

such that the inequality

0

<

|

l

-

j

k

|

<

1

k

n

holds. All Liouville numbers are

transcendental numbers

, but not all transcendental numbers are Liouville numbers. The first example given by Joseph Liouville was a number of the form

∑

i

=

1

∞

1

b

i

!

,

with an integer

b

>

1

, specifically

b

=

10

(the resulting number is now called

Liouville’s constant

and is listed in A012245 of Sloane’s OEIS). In base

b

, a number of this form has a

representation

beginning 0.110001000000000000000001000… where the

i

th instance of the digit 1 is separated from the previous by

i

!

-

(

i

-

1

)

!

-

1

instances of the digit 0.

The set of Liouville numbers is small in measure, having measure 0; in the measure-theoretic setting almost all numbers are not Liouville numbers. However, the set of Liouville numbers is topologically big, as it is residual and in the topological setting almost all numbers are Liouville numbers.

Title

Liouville number

Canonical name

LiouvilleNumber

Date of creation

2013-03-22 17:18:32

Last modified on

2013-03-22 17:18:32

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11J81

Defines

Liouville’s constant

Generated on Fri Feb 9 15:35:40 2018 by

LaTeXML
Prouhet-Thue-Morse constant

Prouhet-Thue-Morse constant

The

Prouhet-Thue-Morse constant

is the number

τ

whose binary expansion is the

Prouhet-Thue-Morse sequence

. That is,

τ

=

∑

i

=

0

∞

t

i

2

i

+

1

=

0.412454033640

⁢

…

where

t

i

is the Prouhet-Thue-Morse sequence

Another

expression

, not in of

t

i

, is

τ

=

1

2

-

1

4

⁢

∏

n

=

0

∞

(

1

-

2

-

2

i

)

The number

τ

has been shown to be transcendental.

Title

Prouhet-Thue-Morse constant

Canonical

name

ProuhetThueMorseConstant

Date of creation

2013-03-22 14:28:23

Last modified on

2013-03-22 14:28:23

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11J81

Classification

msc 11B85

Related topic

ProuhetThueMorseSequence

Defines

Thue-Morse constant

Generated on Fri Feb 9 15:35:41 2018 by

LaTeXML
Roth’s theorem

Roth’s theorem

The following

theorem

is due to Klaus Roth and it is a

generalization

of a previous theorem of Liouville (see

Liouville approximation theorem

). Roth was awarded the

Fields Medal

for his work on the geometry of numbers. W. M. Schmidt generalized the result even further. The result is widely used to prove that a certain number is

transcendental

. Here, for a

rational number

t

in reduced form, the

denominator

of

t

is denoted by

d

⁢

(

t

)

.

Theorem 1

.

For any

algebraic number

α

and for any

ϵ

>

0

there are only finitely many rational numbers

t

with:

|

α

-

t

|

<

1

d

⁢

(

t

)

2

+

ϵ

.

In other words, the equation:

|

α

-

p

q

|

<

1

q

2

+

ϵ

has only finitely many solutions with

p

∈

Z

and

q

∈

Z

+

.

Title

Roth’s theorem

Canonical name

RothsTheorem

Date of creation

2013-03-22 15:02:23

Last modified on

2013-03-22 15:02:23

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

7

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11J81

Classification

msc 11J68

Related topic

ExampleOfTranscendentalNumber

Generated on Fri Feb 9 15:35:43 2018 by

LaTeXML
All unnatural square roots are irrational

All unnatural square roots are irrational

Theorem

:

If

n

is a

natural number

and

n

2

is not whole, then

n

2

must be

irrational

.

Proof

Ad absurdum:
Assume there exists a natural number

n

that

n

2

is not whole, but is

rational

.

Therefore

n

2

can be notated as an irreducible fraction:

m

d

Now break the

numerator

and denominator into their

prime factors

:

n

2

=

m

d

=

m

1

×

m

2

×

…

×

m

k

d

1

×

…

×

d

l

Because the fraction is

irreducible

, none of the factors can cancel each other out.

For any

i

and

j

,

m

i

≠

d

j

.

Now look at

n

:

n

=

m

1

2

×

m

2

2

×

…

×

m

k

2

d

1

2

×

…

×

d

l

2

Because

n

is a natural number, all the denominator factors are supposed to cancel out,

but this is impossible because for any

i

and

j

,

m

i

≠

d

j

.

Therefore

n

2

must be irrational.

Unfortunately this means that a (non-integer) fraction can never become whole by simply squaring, cubing, etc.

I call this unsatisfying fact my ”Greenfield Lemma”.

Title

All unnatural

square roots

are irrational

Canonical name

AllUnnaturalSquareRootsAreIrrational

Date of creation

2013-03-22 17:37:03

Last modified on

2013-03-22 17:37:03

Owner

ubershmekel (18723)

Last modified by

ubershmekel (18723)

Numerical id

12

Author

ubershmekel (18723)

Entry type

Theorem

Classification

msc 11J82

Classification

msc 11J72

Synonym

The square of a fraction is always a fraction

Related topic

RationalBriggsianLogarithmsOfIntegers

Generated on Fri Feb 9 15:35:48 2018 by

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e is irrational

e is irrational

From the

Taylor series

for

e

x

we know the following equation:

e

=

∑

k

=

0

∞

1

k

!

.

(1)

Now let us assume that

e

is

rational

. This would there are two

natural numbers

a

and

b

, such that:

e

=

a

b

.

This yields:

b

!

⁢

e

∈

ℕ

.

Now we can write

e

using (1):

b

!

⁢

e

=

b

!

⁢

∑

k

=

0

∞

1

k

!

.

This can also be written:

b

!

⁢

e

=

∑

k

=

0

b

b

!

k

!

+

∑

k

=

b

+

1

∞

b

!

k

!

.

The first sum is obviously a natural number, and thus

∑

k

=

b

+

1

∞

b

!

k

!

must also be . Now we see:

∑

k

=

b

+

1

∞

b

!

k

!

=

1

b

+

1

+

1

(

b

+

1

)

⁢

(

b

+

2

)

+

…

<

∑

k

=

1

∞

(

1

b

+

1

)

k

=

1

b

.

Since

1

b

≤

1

we conclude:

0

<

∑

k

=

b

+

1

∞

b

!

k

!

<

1

.

We have also seen that this is an integer, but there is no integer between 0 and 1. So there cannot exist two natural numbers

a

and

b

such that

e

=

a

b

, so

e

is irrational.

Title

e is irrational

Canonical name

EIsIrrational

Date of creation

2013-03-22 12:33:02

Last modified on

2013-03-22 12:33:02

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

13

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 11J82

Classification

msc 11J72

Related topic

ErIsIrrationalForRinmathbbQsetminus0

Related topic

NaturalLogBase

Generated on Fri Feb 9 15:35:50 2018 by

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e is irrational

e is irrational

We have the series

e

-

1

=

∑

k

=

0

∞

(

-

1

)

k

k

!

Note that this is an

alternating series

and that the magnitudes of the
terms decrease. Hence, for every integer

n

>

0

, we have the bound

0

<

|

∑

k

=

0

n

(

-

1

)

k

k

!

-

e

-

1

|

<

1

(

n

+

1

)

!

,

by the Leibniz’ estimate for alternating
series (

http://

planetmath

.org/LeibnizEstimateForAlternatingSeries

).  Assume
that

e

=

n

/

m

, where

m

and

n

are integers and

n

>

0

.
Then we would have

0

<

|

∑

k

=

0

n

(

-

1

)

k

k

!

-

m

n

|

<

1

(

n

+

1

)

!

.

Multiplying both sides by

n

!

, this would imply

0

<

|

∑

k

=

0

n

(

-

1

)

k

⁢

n

!

k

!

-

m

⁢

(

n

-

1

)

!

|

<

1

n

+

1

,

which is a

contradiction

because every term in the sum is an integer,
but there are no integers between

0

and

1

/

(

n

+

1

)

.

Title

e is irrational

Canonical name

EIsIrrational1

Date of creation

2013-03-22 17:02:32

Last modified on

2013-03-22 17:02:32

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 11J82

Classification

msc 11J72

Related topic

LeibnizEstimateForAlternatingSeries

Generated on Fri Feb 9 15:35:53 2018 by

LaTeXML
example of transcendental number

example of transcendental number

The following is a classical application of Liouville’s

approximation theorem

. For completeness, we state Liouville’s result here:

Theorem 1

.

For any

algebraic number

α

with degree

m

>

1

, there exists a

constant

c

=

c

⁢

(

α

)

>

0

such that:

|

α

-

p

q

|

>

c

q

m

for all

rationals

p

/

q

(with

q

>

0

).

Next we use the

theorem

to construct a

transcendental number

.

Corollary 1

.

The

real number

ψ

=

∑

n

=

1

∞

1

10

n

!

=

0.1100010

⁢

…

is

transcendental

.

Proof.

Clearly, the number

ψ

is well defined, i.e. the series

converges

. Indeed,

1

10

n

!

<

1

10

n

and

∑

n

=

1

∞

10

-

n

=

1

/

9

. Thus, by the comparison test, the series converges and

0

<

ψ

<

1

/

9

.

Suppose, for a

contradiction

, that

ψ

is

algebraic

of degree

m

. We will construct infinitely many rationals

p

/

q

such that

|

ψ

-

p

q

|

<

c

q

m

where

c

=

c

⁢

(

ψ

)

is the constant given by the theorem above. Let

k

∈

ℕ

be such that

1

/

2

k

<

c

. Then, in fact, we will show that there are infinitely many rationals

p

/

q

with

q

≥

2

such that

|

ψ

-

p

q

|

<

1

q

m

+

k

<

1

2

k

⋅

1

q

m

<

c

q

m

For all

j

>

k

+

m

we define a rational number

p

j

/

q

j

by:

p

j

=

10

j

!

⁢

∑

n

=

1

j

10

-

n

!

,

q

j

=

10

j

!

then

p

j

and

q

j

are relatively

prime integers

and we have:

|

ψ

-

p

j

q

j

|

=

∑

n

=

j

+

1

∞

1

10

n

!

<

1

10

(

j

+

1

)

!

⁢

(

1

+

1

10

+

1

10

2

+

…

)

=

10

/

9

⋅

1

q

j

(

j

+

1

)

<

1

q

j

j

<

1

q

j

(

k

+

m

)

where in the last

inequality

we have used the fact that

j

>

k

+

m

. Therefore, all rationals

{

p

j

/

q

j

}

j

=

k

+

m

+

1

∞

satisfy

the desired inequality, which leads to the contradiction with the theorem above. Thus

ψ

cannot be algebraic and it must be transcendental.
∎

Many other

similar

transcendental numbers can be constructed in this fashion.

Title

example of transcendental number

Canonical name

ExampleOfTranscendentalNumber

Date of creation

2013-03-22 15:02:45

Last modified on

2013-03-22 15:02:45

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

7

Author

alozano (2414)

Entry type

Example

Classification

msc 11J82

Classification

msc 11J81

Related topic

LiouvillesTheorem

Related topic

RothsTheorem

Generated on Fri Feb 9 15:35:57 2018 by

LaTeXML
if a^n is irrational then a is irrational

if

a

n

is irrational then

a

is irrational

Theorem

.

If

a

be a

real number

and

n

is an integer such that

a

n

is irrational, then

a

is irrational.

Proof.

We show this by way of

contrapositive

. In other words, we show that, if

a

is

rational

, then

a

n

is rational.

Let

a

be rational. Then there exist integers

b

and

c

with

c

≠

0

such that

a

=

b

c

. Thus,

a

n

=

b

n

c

n

, which is a

rational number

.
∎

Note that the

converse

is not true. For example,

2

is irrational and

(

2

)

2

=

2

is rational.

Title

if

a

n

is irrational then

a

is irrational

Canonical name

IfAnIsIrrationalThenaIsIrrational

Date of creation

2013-03-22 14:18:50

Last modified on

2013-03-22 14:18:50

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11J82

Classification

msc 11J72

Generated on Fri Feb 9 15:35:59 2018 by

LaTeXML
irrational

irrational

An

irrational number

is a real number which cannot be represented as a ratio of two

integers

. That is, if

x

is irrational, then

x

≠

a

b

with

a

,

b

∈

ℤ

and

b

≠

0

.

Examples

1.

2

p

is irrational for

p

=

2

,

3

,

…

,

2.

π

,

e

, and

2

p

for

p

=

2

,

3

,

…

,
are irrational,

3.

It is not known whether Euler’s constant is

rational

or irrational.

Properties

1.

It

a

is a real number and

a

n

is irrational for some

n

=

2

,

3

,

…

,
then

a

is irrational (proof (

http://planetmath.org/IfAnIsIrrationalThenAIsIrrational

)).

2.

The sum,

difference

, product, and

quotient

(when defined) of two numbers,
one rational and another irrational, is irrational.
(proof (

http://planetmath.org/RationalAndIrrational

)).

Title

irrational

Canonical

name

Irrational

Date of creation

2013-03-22 11:55:59

Last modified on

2013-03-22 11:55:59

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Definition

Classification

msc 11J82

Classification

msc 11J72

Synonym

irrational number

Related topic

TranscedentalNumber

Related topic

AlgebraicNumber

Related topic

Integer

Related topic

LindemannWeierstrassTheorem

Related topic

GelfondsTheorem

Related topic

ProofThatTheRationalsAreCountable

Generated on Fri Feb 9 15:36:01 2018 by

LaTeXML
irrationality measure

irrationality measure

Let

α

∉

ℚ

.
Let

M

⁢

(

α

)

=

{

μ

>

0

∣

∃

q

0

=

q

0

⁢

(

α

,

μ

)

>

0

⁢

such that

⁢

|

α

-

p

q

|

>

1

q

μ

∀

p

,

q

∈

ℤ

,

q

>

q

0

}

.

The

irrationality measure

of

α

,
denoted by

μ

⁢

(

α

)

, is defined by

μ

⁢

(

α

)

=

inf

⁡

M

⁢

(

α

)

.

If

M

⁢

(

α

)

=

∅

, we set

μ

⁢

(

α

)

=

∞

.

This

definition

is (loosely) a measure of the extent to which

α

can be approximated by

rational numbers

. Of course, by the fact that

ℚ

is dense in

ℝ

, we can make arbitrarily good approximations to real numbers by rationals. Thus this definition was made to represent a stronger statement: it is the ability of rational numbers to approximate

α

given a fixed growth bound on the

denominators

of those rational numbers.

By the Dirichlet’s Lemma,

μ

⁢

(

α

)

≥

2

.
Roth

[

6

,

7

]

proved in 1955 that

μ

⁢

(

α

)

=

2

for every

algebraic

real number.
It is well known also
that

μ

⁢

(

e

)

=

2

. For almost all real numbers
the irrationality measure is 2.
However, for special

constants

, only some

upper bounds

are known:

Constant

Upper bound

Reference

π

8.0161

Hata (1993)

[

3

]

π

/

3

4.6016

Hata (1993)

[

3

]

e

2

Davis (1978)

[

1

]

π

2

5.4413

Rhin and Viola (1996)

[

4

]

log

⁡

2

3.8914

Rukhadze (1987)

[

8

]

, Hata (1990)

[

2

]

ζ

⁢

(

3

)

5.5139

Rhin and Viola (2001)

[

5

]

It is worth noting that the last column of the above table is simply a list of references, not a

collection

of discoverers. For example that fact that the irrationality measure of

e

is 2 was known to Euler.

References

1

Davis, C.S.,

‘

Rational approximations

to

e

’

,
J. Austral. Math. Soc. Ser. A

25

(1978), 497–502.

2

Hata, M.

‘Legendre Type

Polynomials

and Irrationality Measures’

, J. reine angew. Math.

407

, 99–125, 1990.

3

Hata, M.,

‘Rational approximations to

π

and some other numbers’

,
Acta Arith.

63

, 335–349 (1993).

4

Rhin, G. and Viola, C.

‘On a

permutation group

related to
zeta(2)

’,
Acta Arith.

77

(1996),
23–56.

5

Rhin, G. and Viola, C.

‘The group

structure

for

ζ

⁢

(

3

)

’

,
Acta Arith.

97

(2001), 269–293.

6

Roth, K.F.,

‘Rational Approximations to

Algebraic Numbers

’

,
Mathematika

2

(1955), 1–20.

7

Roth, K.F.

‘Corrigendum to
’Rational Approximations to Algebraic Numbers”

Mathematika

2

(1955), 168.

8

Rukhadze, E.A.

‘A Lower Bound for the Rational Approximation of by Rational Numbers’

Vestnik Moskov Univ. Ser. I Math. Mekh.,

6

(1987), 25-29 and 97.

Title

irrationality measure

Canonical

name

IrrationalityMeasure

Date of creation

2013-03-22 14:12:22

Last modified on

2013-03-22 14:12:22

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11J82

Related topic

DirichletsApproximationTheorem

Generated on Fri Feb 9 15:36:03 2018 by

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transcendental number

transcendental number

A

transcendental number

is a

complex number

that is not an

algebraic number

.
That is, it is a complex number that is

transcendental

over

ℚ

(or, equivalently, over

ℤ

).

Well known transcendental numbers include

π

and

e

(the base of

natural logarithms

).

Cantor showed that, in a sense,
“almost all” complex numbers are transcendental:
there are uncountably many complex numbers, but
only countably many algebraic numbers (

http://planetmath.org/AlgebraicNumbersAreCountable

).

Title

transcendental number

Canonical name

TranscendentalNumber

Date of creation

2013-03-22 11:55:54

Last modified on

2013-03-22 11:55:54

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Definition

Classification

msc 11J82

Classification

msc 11J81

Related topic

Pi

Related topic

Irrational

Generated on Fri Feb 9 15:36:05 2018 by

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multiplicatively independent

multiplicatively independent

A set

X

of nonzero

complex numbers

is said to be

multiplicatively independent

iff every equation

x

1

ν

1

⁢

x

2

ν

2

⁢

⋯

⁢

x

n

ν

n

=

1

with

x

1

,

x

2

,

…

,

x

n

∈

X

and

ν

1

,

ν

2

,

…

,

ν

n

∈

ℤ

implies that

ν

1

=

ν

2

=

…

=

ν

n

=

0

.

For example, the set of

prime numbers

is multiplicatively independent, by the

fundamental theorem of arithmetics

.

Any

algebraically independent

set is also multiplicatively independent.

Evidently,

{

x

1

,

x

2

,

…

,

x

n

}

is multiplicatively
independent if and only if the numbers

log

⁡

x

1

,

log

⁡

x

2

, …,

log

⁡

x

n

are

linearly independent

over

ℚ

.  Thus the Schanuel’s conjecture may be formulated as the

Conjecture.

If

{

x

1

,

x

2

,

…

,

x

n

}

is multiplicatively independent, then the

transcendence degree

of the set

{

x

1

,

x

2

,

…

,

x

n

,

log

⁡

x

1

,

log

⁡

x

2

,

…

,

log

⁡

x

n

}

is at least

n

.

References

1

Diego Marques & Jonathan Sondow

:

Schanuel’s conjecture and

algebraic

powers

z

w

and

w

z

with

z

and

w

transcendental

(2011). Available

http://arxiv.org/pdf/1010.6216.pdf

here.

Title

multiplicatively independent

Canonical name

MultiplicativelyIndependent

Date of creation

2013-03-22 19:36:03

Last modified on

2013-03-22 19:36:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 11J85

Classification

msc 12F05

Generated on Fri Feb 9 15:36:08 2018 by

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Goormaghtigh conjecture

Goormaghtigh conjecture

For

integers

x

,

y

,

m

and

n

satisfying

x

>

1

,

y

>

1

, and

n

>

m

>

2

, the equation

x

m

-

1

x

-

1

=

y

n

-

1

y

-

1

has only the solutions

(

x

,

y

,

m

,

n

)

=

(

5

,

2

,

3

,

5

)

and

(

90

,

2

,

3

,

13

)

.

See the following paper for some progress’ on the

conjecture

:

M. Le, Exceptional solutions of the exponential

Diophantine equation

(

x

3

-

1

)

/

(

x

-

1

)

=

(

y

n

-

1

)

/

(

y

-

1

)

. J. Reine Angew. Math.

543

(2002) 187-192.

See

Section

7 of following paper in which the solution of a certain case of the
conjecture (given in the
latter qouted paper) is used to solve a
problem in group theory concerning the non-cyclic graph of a group. Also see

Proposition

7.6 of
the folloiwng paper for an slightly special case of the conjecture which its
solotion has some
applications in group theory.

Alireza Abdollahi and A. Mohammadi Hassanabadi, Non-cyclic graph of a group,
Communications in

Algebra

,

35

(2007) 2057-2081.

Title

Goormaghtigh conjecture

Canonical

name

GoormaghtighConjecture

Date of creation

2013-03-22 15:18:55

Last modified on

2013-03-22 15:18:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Conjecture

Classification

msc 11J86

Classification

msc 11J61

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