aarjaneiro/planetmath huggingface dataset:
cube root

cube root

The

cube root

of a

real number

x

, written as

x

3

, is the real number

y

such that

y

3

=

x

.
Equivalently,

x

3

3

=

x

. Or,

x

3

⁢

x

3

⁢

x

3

=

x

. The cube root notation is actually an alternative to exponentiation. That is,

x

3

=

x

1

3

.

Properties

:

•

The cube root operation of an exponentiation has the following property:

x

n

3

=

x

3

n

.

•

The cube root operation is distributive for

multiplication

and

division

, but not for

addition

and

subtraction

. That is,

x

⁢

y

3

=

x

3

⁢

y

3

, and

x

y

3

=

x

3

y

3

.

•

However, in general, the cube root operation is not distributive for addition and substraction. That is,

x

+

y

3

≠

x

3

+

y

3

and

x

-

y

3

≠

x

3

-

y

3

.

•

The cube root is a special case of the general

nth root

.

•

The cube root is a

continuous mapping

from

ℝ

→

ℝ

.

•

The cube root function from

ℝ

→

ℝ

defined as

f

⁢

(

x

)

=

x

3

is an

odd function

.

Examples:

1.

-

8

3

=

-

2

because

(

-

2

)

3

=

(

-

2

)

×

(

-

2

)

×

(

-

2

)

=

-

8

.

2.

x

3

+

3

⁢

x

2

+

3

⁢

x

+

1

3

=

x

+

1

because

(

x

+

1

)

3

=

(

x

+

1

)

⁢

(

x

+

1

)

⁢

(

x

+

1

)

=

(

x

2

+

2

⁢

x

+

1

)

⁢

(

x

+

1

)

=

x

3

+

3

⁢

x

2

+

3

⁢

x

+

1

.

3.

x

3

⁢

y

3

3

=

x

⁢

y

because

(

x

⁢

y

)

3

=

x

⁢

y

×

x

⁢

y

×

x

⁢

y

=

x

3

⁢

y

3

.

4.

8

125

3

=

2

5

because

(

2

5

)

3

=

2

3

5

3

=

8

125

.

Title

cube root

Canonical

name

CubeRoot

Date of creation

2013-03-22 11:57:22

Last modified on

2013-03-22 11:57:22

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

12

Author

Daume (40)

Entry type

Definition

Classification

msc 11-00

Related topic

NthRoot

Related topic

SquareRoot

Related topic

RationalNumber

Related topic

IrrationalNumber

Related topic

RealNumber

Related topic

ComplexNumber

Related topic

CubeOfANumber

Generated on Fri Feb 9 15:06:45 2018 by

LaTeXML
cyclotomic extension

cyclotomic extension

Definition

.

Let

K

be a field and let

K

¯

be a fixed

algebraic closure

of

K

. A

cyclotomic extension

of

K

is an

extension field

of the form

K

⁢

(

ζ

)

where

ζ

∈

K

¯

is a

root of unity

.

Title

cyclotomic extension

Canonical name

CyclotomicExtension

Date of creation

2013-03-22 15:05:17

Last modified on

2013-03-22 15:05:17

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 11-00

Classification

msc 11R18

Related topic

KroneckerWeberTheorem

Related topic

CyclotomicField

Generated on Fri Feb 9 15:06:47 2018 by

LaTeXML
derivation of Euler phi-function

derivation of Euler phi-function

In this “proof” we will construct the solution for the Euler phi-function,

ϕ

⁢

(

n

)

=

n

⁢

∏

p

|

n

(

1

-

1

n

)

.

We will do this for the

natural number

n

>

0

.
Keep in mind that

gcd

⁡

(

a

,

n

)

=

1

⟺

a

is not

divisible

by

p

for all primes

p

dividing

n

.

Let

n

≥

2

and

p

1

,

p

2

,

⋯

,

p

r

be all

prime divisors

of n.
Let

N

=

{

a

∣

0

≤

a

<

n

,

gcd

⁡

(

a

,

n

)

=

1

}

and

A

i

:=

{

a

∣

0

≤

a

⁢

<

n

,

p

i

|

⁢

a

}

.
If

J

⊂

{

1

,

2

,

⋯

,

r

}

than

p

J

:=

∏

j

∈

J

p

i

.

Thus,

#

(

A

J

)

=

#

(

⋂

j

∈

J

A

j

)

=

#

(

{

a

∈

A

:

p

J

|

a

}

)

=

n

p

J

Using inclusion-exclusion,

ϕ

⁢

(

n

)

=

#

⁢

(

N

)

=

∑

J

⊂

{

1

,

2

,

⋯

,

r

}

(

-

1

)

#

⁢

(

J

)

⁢

#

⁢

(

A

J

)

=

∑

J

⊂

{

1

,

2

,

⋯

,

r

}

(

-

1

)

#

⁢

(

J

)

⁢

n

p

J

=

n

⁢

∑

J

⊂

{

1

,

2

,

⋯

,

r

}

(

-

1

)

#

⁢

(

J

)

⁢

1

p

J

=

n

⁢

(

1

-

(

1

p

1

+

1

p

2

+

⋯

+

1

p

r

)

+

⋯

⁢

(

-

1

)

r

⁢

1

p

1

⁢

p

2

⁢

⋯

⁢

p

r

)

=

n

⁢

∏

p

|

n

(

1

-

1

p

)

.

□

Title

derivation of Euler phi-function

Canonical name

DerivationOfEulerPhifunction

Date of creation

2013-03-22 17:42:52

Last modified on

2013-03-22 17:42:52

Owner

jwaixs (18148)

Last modified by

jwaixs (18148)

Numerical id

22

Author

jwaixs (18148)

Entry type

Derivation

Classification

msc 11-00

Generated on Fri Feb 9 15:06:49 2018 by

LaTeXML
even number

even number

Definition

Suppose

k

is an

integer

.
If there exists an integer

r

such that

k

=

2

⁢

r

+

1

, then

k

is an

odd number

.
If there exists an integer

r

such that

k

=

2

⁢

r

, then

k

is an

even number

.

The concept of even and odd numbers are most easily understood in
the binary base. Then the above definition simply that even numbers end
with a

0

, and odd numbers end with a

1

.

0.0.1

Properties

1.

Every integer is either even or . This can be proven
using induction, or using the

fundamental theorem of arithmetic

.

2.

An integer

k

is even () if and only if

k

2

is even ().

Title

even number

Canonical name

EvenNumber

Date of creation

2013-03-22 13:56:29

Last modified on

2013-03-22 13:56:29

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11-00

Classification

msc 03-00

Related topic

NumberOdd

Defines

odd number

Defines

even integer

Defines

odd integer

Defines

even

Defines

odd

Generated on Fri Feb 9 15:06:51 2018 by

LaTeXML
existence and uniqueness of the gcd of two integers

existence and uniqueness of the gcd of two integers

Theorem

.

Given two

integers

, at least one different from zero, there exists a unique

natural number

satisfying the definition of the

greatest common divisor

.

Proof.

Let

a

,

b

∈

ℤ

, where at least one of

a

,

b

is nonzero. First we show

existence

. Define

S

=

{

m

⁢

a

+

n

⁢

b

:

m

,

n

∈

ℤ

⁢

and

⁢

m

⁢

a

+

n

⁢

b

>

0

}

. Now clearly

S

is a subset of natural numbers, and

S

is also nonempty, for depending upon the signs of

a

and

b

, we may take

m

=

n

=

±

1

to have

m

⁢

a

+

n

⁢

b

∈

S

. So, by the

well-ordering principle for natural numbers

,

S

has a smallest

element

which we denote

g

.
Note that, by construction,

g

=

m

0

⁢

a

+

n

0

⁢

b

for some

m

0

,

n

0

∈

ℤ

. We will show that

g

is a greatest common divisor of

a

and

b

. Suppose first that

g

∤

a

. Then, by the

division algorithm for integers

, there exist unique

q

,

r

∈

ℤ

, where

0

≤

r

<

g

, such that

a

=

q

⁢

g

+

r

. By

assumption

,

g

∤

a

, so we have

0

<

r

=

a

-

q

⁢

g

=

a

-

q

⁢

(

m

0

⁢

a

+

n

0

⁢

b

)

=

a

-

q

⁢

m

0

⁢

a

-

q

⁢

n

0

⁢

b

=

(

1

-

q

⁢

m

0

)

⁢

a

-

q

⁢

n

0

⁢

b

<

g

⁢

.

But then

r

is an element of

S

strictly less than

g

, contrary to assumption. Thus it must be that

g

∣

a

. Similarly it can be shown that

g

∣

b

. Now suppose

h

∈

ℕ

is a

divisor

of both

a

and

b

. Then there exist

k

,

l

∈

ℤ

such that

a

=

k

⁢

h

and

b

=

l

⁢

h

, and we have

g

=

m

0

⁢

a

+

n

0

⁢

b

=

m

0

⁢

k

⁢

h

+

n

0

⁢

l

⁢

h

=

(

m

0

⁢

k

+

n

0

⁢

l

)

⁢

h

⁢

,

so

h

∣

g

.
Thus

g

is a greatest common divisor of

a

and

b

. To see that

g

is unique, suppose that

g

′

∈

ℕ

is also a greatest common divisor of

a

and

b

. Then we have

g

′

∣

g

and

g

∣

g

′

, whence

g

=

±

g

′

, and since

g

,

g

′

>

0

,

g

=

g

′

.
∎

Title

existence and uniqueness of the gcd of two integers

Canonical name

ExistenceAndUniquenessOfTheGcdOfTwoIntegers

Date of creation

2013-03-22 16:28:44

Last modified on

2013-03-22 16:28:44

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11-00

Related topic

GreatestCommonDivisor

Related topic

DivisibilityInRings

Related topic

DivisionAlgorithmForIntegers

Related topic

WellOrderingPrinciple

Generated on Fri Feb 9 15:06:55 2018 by

LaTeXML
existence of primitive roots for powers of an odd prime

existence of primitive roots for powers of an odd prime

The following theorem gives a way of finding a

primitive root

for

p

k

, for an

odd prime

p

and

k

≥

1

, given a primitive root of

p

. Recall that every prime has a primitive root.

Theorem

.

Suppose that

p

is an odd prime. Then

p

k

also has a primitive root, for all

k

≥

1

. Moreover:

1.

If

g

is a primitive root of

p

and

g

p

-

1

≠

1

mod

p

2

then

g

is a primitive root of

p

2

. Otherwise, if

g

p

-

1

≡

1

mod

p

2

then

g

+

p

is a primitive root of

p

2

.

2.

If

k

≥

2

and

h

is a primitive root of

p

k

then

h

is a primitive root of

p

k

+

1

.

Title

existence of primitive roots for powers of an odd prime

Canonical name

ExistenceOfPrimitiveRootsForPowersOfAnOddPrime

Date of creation

2013-03-22 16:21:01

Last modified on

2013-03-22 16:21:01

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11-00

Related topic

EveryPrimeHasAPrimitiveRoot

Generated on Fri Feb 9 15:06:57 2018 by

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extracting every n^th term of a series

extracting every

n

th

term of a series

Roots of unity

can be used to extract every

n

th

term of a series. This method is due to Simpson [1759].

Theorem

.

Let

ω

=

e

2

⁢

π

⁢

i

/

k

be a primitive

k

th

root of unity. If

f

⁢

(

x

)

=

∑

j

=

0

∞

a

j

⁢

x

j

and

n

≢

0

(

mod

k

)

, then

∑

j

=

0

∞

a

k

⁢

j

+

n

⁢

x

k

⁢

j

+

n

=

1

k

⁢

∑

j

=

0

k

-

1

ω

-

j

⁢

n

⁢

f

⁢

(

ω

j

⁢

x

)

Proof.

This is a

consequence

of the fact that

∑

j

=

0

k

-

1

ω

j

⁢

m

=

0

for

m

≢

0

(

mod

k

)

.

Consider the term involving

x

r

on the right-hand side. It is

1

k

⁢

∑

j

=

0

k

-

1

ω

-

j

⁢

n

⁢

a

r

⁢

ω

j

⁢

r

⁢

x

r

=

1

k

⁢

a

r

⁢

x

r

⁢

∑

j

=

0

k

-

1

ω

j

⁢

(

r

-

n

)

If

r

≢

n

(

mod

k

)

, the sum is zero. So the term involving

x

r

is zero unless

r

≡

n

(

mod

k

)

, in which case it is

a

r

⁢

x

r

since each element of the sum is

1

.

Note that this method is a

generalization

of the commonly known trick for extracting alternate terms of a series:

1

2

⁢

(

f

⁢

(

x

)

-

f

⁢

(

-

x

)

)

produces the odd terms of

f

.

Title

extracting every

n

th

term of a series

Canonical name

ExtractingEveryNmathrmthTermOfASeries

Date of creation

2013-03-22 16:23:34

Last modified on

2013-03-22 16:23:34

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

10

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11-00

Generated on Fri Feb 9 15:06:59 2018 by

LaTeXML
Fermat’s little theorem

Fermat’s little theorem

Theorem

(Fermat’s little theorem).

If

a

,

p

∈

Z

with

p

a prime and

p

∤

a

, then

a

p

-

1

≡

1

(

mod

p

)

.

If we take away the condition that

p

∤

a

, then we have the

congruence relation

a

p

≡

a

(

mod

p

)

instead.

Remarks

.

•

Although Fermat first noticed this property, he never actually proved it. There are several different ways to directly prove this

theorem

, but it is really just a corollary of the

Euler theorem

.

•

More generally, this is a statement about

finite fields

: if

K

is a finite field of order

q

, then

a

q

-

1

=

1

for all

0

≠

a

∈

K

. More succinctly, the

group of units

in a finite field is cyclic. If

q

is prime, then we have Fermat’s little theorem.

•

While it is true that

p

prime implies the congruence relation above, the

converse

is false (as hypothesized by ancient Chinese mathematicians). A well-known example of this is provided by setting

a

=

2

and

p

=

341

=

11

×

31

. It is easy to verify that

2

341

≡

2

(

mod

341

)

. A

positive integer

p

satisfying

a

p

-

1

≡

1

(

mod

p

)

is known as a

pseudoprime

of base

a

.

Fermat little theorem

says that every prime is a pseudoprime of any base not

divisible

by the prime.

References

1

H. Stark,

An

Introduction

to

Number Theory

. The MIT Press (1978)

Title

Fermat’s little theorem

Canonical name

FermatsLittleTheorem

Date of creation

2013-03-22 11:45:08

Last modified on

2013-03-22 11:45:08

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 11-00

Classification

msc 16E30

Synonym

Fermat’s theorem

Related topic

EulerFermatTheorem

Related topic

ProofOfEulerFermatTheoremUsingLagrangesTheorem

Related topic

FermatsTheoremProof

Related topic

PolynomialCongruence

Generated on Fri Feb 9 15:07:01 2018 by

LaTeXML
Fermat’s theorem proof

Fermat’s theorem proof

Consider the sequence

a

,

2

⁢

a

,

…

,

(

p

-

1

)

⁢

a

.

They are all different (modulo

p

), because if

m

⁢

a

=

n

⁢

a

with

1

≤

m

<

n

≤

p

-

1

then

0

=

a

⁢

(

m

-

n

)

, and since

p

∤

a

, we get

p

∣

(

m

-

n

)

,  which is impossible.

Now, since all these numbers are different, the set

{

a

,

2

⁢

a

,

3

⁢

a

,

…

,

(

p

-

1

)

⁢

a

}

will have the

p

-

1

possible congruence classes (although not necessarily in the same order) and therefore

a

⋅

2

⁢

a

⋅

3

⁢

a

⁢

⋯

⁢

(

p

-

1

)

⁢

a

≡

(

p

-

1

)

!

⁢

a

p

-

1

≡

(

p

-

1

)

!

(

mod

p

)

and using

gcd

⁡

(

(

p

-

1

)

!

,

p

)

=

1

we get

a

p

-

1

≡

1

(

mod

p

)

.

Title

Fermat’s theorem proof

Canonical name

FermatsTheoremProof

Date of creation

2013-03-22 11:46:10

Last modified on

2013-03-22 11:46:10

Owner

drini (3)

Last modified by

drini (3)

Numerical id

11

Author

drini (3)

Entry type

Proof

Classification

msc 11-00

Classification

msc 37B55

Related topic

EulerFermatTheorem

Related topic

FermatsLittleTheorem

Related topic

ProofOfEulerFermatTheoremUsingLagrangesTheorem

Related topic

FermatsLittleTheoremProofInductive

Generated on Fri Feb 9 15:07:03 2018 by

LaTeXML
fractional part

fractional part

The

fractional part

of a nonnegative

real number

is the part of the number that appears after the decimal . For example, the fractional part of

7

3

is

1

3

.

To be more precise, for

x

∈

ℝ

with

x

≥

0

, the fractional part of

x

, denoted as

{

x

}

, is given by

{

x

}

=

x

-

[

x

]

, where

[

x

]

denotes the

integer part

of

x

.

The name “fractional part” is somewhat of a misnomer: To the novice, the name may seem to imply that the result must be a

fraction

(and therefore

rational

), which is not the case. For example,

{

π

}

=

π

-

3

, which is not rational.

Title

fractional part

Canonical

name

FractionalPart

Date of creation

2013-03-22 16:14:13

Last modified on

2013-03-22 16:14:13

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11-00

Classification

msc 26A09

Related topic

IntegerPart

Generated on Fri Feb 9 15:07:05 2018 by

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geometric mean

geometric mean

Geometric Mean

.

If

a

1

,

a

2

,

…

,

a

n

are

real numbers

, we define their

geometric mean

as

G

.

M

.

=

a

1

⁢

a

2

⁢

⋯

⁢

a

n

n

(We usually require the numbers to be non

negative

so the mean always exists.)

Title

geometric mean

Canonical name

GeometricMean

Date of creation

2013-03-22 11:50:46

Last modified on

2013-03-22 11:50:46

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 11-00

Classification

msc 44A20

Classification

msc 33E20

Classification

msc 30D15

Related topic

ArithmeticMean

Related topic

GeneralMeansInequality

Related topic

WeightedPowerMean

Related topic

PowerMean

Related topic

ArithmeticGeometricMeansInequality

Related topic

ProofOfArithmeticGeome

Related topic

RootMeanSquare3

Related topic

ProofOfGeneralMeansInequality

Related topic

DerivationOfZerothWeightedPowerMean

Related topic

ProofOfArithmeticGeometricHarmonicMeansI

Generated on Fri Feb 9 15:07:08 2018 by

LaTeXML
Goldbach representations for even 5<n<101

Goldbach representations for even

5

<

n

<

101

Christian Goldbach conjectured that all sufficiently large

even integers

can be represented as the sum of two

odd primes

. This has been checked for all even integers up to

10

18

. The table below gives Goldbach

representations

for the

integers

from 8 to 100.

6

3 + 3

8

5 + 3

10

7 + 3 = 5 + 5

12

7 + 5

14

11 + 3 = 7 + 7

16

13 + 3 = 11 + 5

18

13 + 5 = 11 + 7

20

17 + 3 = 13 + 7

22

19 + 3 = 17 + 5 = 11 + 11

24

19 + 5 = 17 + 7 = 13 + 11

26

23 + 3 = 19 + 7 = 13 + 13

28

23 + 5 = 17 + 11

30

23 + 7 = 19 + 11 = 17 + 13

32

29 + 3 = 19 + 13

34

31 + 3 = 29 + 5 = 23 + 11 = 17 + 17

36

31 + 5 = 29 + 7 = 23 + 13 = 19 + 17

38

31 + 7 = 19 + 19

40

37 + 3 = 29 + 11 = 23 + 17

42

37 + 5 = 31 + 11 = 29 + 13 = 23 + 19

44

41 + 3 = 37 + 7 = 31 + 13

46

43 + 3 = 41 + 5 = 29 + 17 = 23 + 23

48

43 + 5 = 41 + 7 = 37 + 11 = 31 + 17 = 29 + 19

50

47 + 3 = 43 + 7 = 37 + 13 = 31 + 19

52

47 + 5 = 41 + 11 = 29 + 23

54

47 + 7 = 43 + 11 = 41 + 13 = 37 + 17 = 31 + 23

56

53 + 3 = 43 + 13 = 37 + 19

58

53 + 5 = 47 + 11 = 41 + 17 = 29 + 29

60

53 + 7 = 47 + 13 = 43 + 17 = 41 + 19 = 37 + 23 = 31 + 29

62

59 + 3 = 43 + 19 = 31 + 31

64

61 + 3 = 59 + 5 = 53 + 11 = 47 + 17 = 41 + 23

66

61 + 5 = 59 + 7 = 53 + 13 = 47 + 19 = 43 + 23 = 37 + 29

68

61 + 7 = 37 + 31

70

67 + 3 = 59 + 11 = 53 + 17 = 47 + 23 = 41 + 29

72

67 + 5 = 61 + 11 = 59 + 13 = 53 + 19 = 43 + 29 = 41 + 31

74

71 + 3 = 67 + 7 = 61 + 13 = 43 + 31 = 37 + 37

76

73 + 3 = 71 + 5 = 59 + 17 = 53 + 23 = 47 + 29

78

73 + 5 = 71 + 7 = 67 + 11 = 61 + 17 = 59 + 19 = 47 + 31 = 41 + 37

80

73 + 7 = 67 + 13 = 61 + 19 = 43 + 37

82

79 + 3 = 71 + 11 = 59 + 23 = 53 + 29 = 41 + 41

84

79 + 5 = 73 + 11 = 71 + 13 = 67 + 17 = 61 + 23 = 53 + 31 = 47 + 37 = 43 + 41

86

83 + 3 = 79 + 7 = 73 + 13 = 67 + 19 = 43 + 43

88

83 + 5 = 71 + 17 = 59 + 29 = 47 + 41

90

83 + 7 = 79 + 11 = 73 + 17 = 71 + 19 = 67 + 23 = 61 + 29 = 59 + 31 = 53 + 37 = 47 + 43

92

89 + 3 = 79 + 13 = 73 + 19 = 61 + 31

94

89 + 5 = 83 + 11 = 71 + 23 = 53 + 41 = 47 + 47

96

89 + 7 = 83 + 13 = 79 + 17 = 73 + 23 = 67 + 29 = 59 + 37 = 53 + 43

98

79 + 19 = 67 + 31 = 61 + 37

100

97 + 3 = 89 + 11 = 83 + 17 = 71 + 29 = 59 + 41 = 53 + 47

Title

Goldbach representations for even

5

<

n

<

101

Canonical name

GoldbachRepresentationsForEven5N101

Date of creation

2013-03-22 17:25:47

Last modified on

2013-03-22 17:25:47

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11-00

Classification

msc 11P32

Generated on Fri Feb 9 15:07:11 2018 by

LaTeXML
googol

googol

Definition

A

googol

equals to the number

10

100

, that is, a one followed by
one

hundred

zeros.
A

googolplex

is
is ten raised to the power of a googol, i.e.,

10

(

10

100

)

.

This is already a huge number. For

instance

,
it is more than the number of atoms in the known

universe

.
A googoplex is even larger. In fact, since a googolplex has
a googol number of zeros in its decimal
representation,
a googoplex has more digits than there are atoms in our universe.
Thus, even if all matter in the universe were at disposal,
it would not be possible to write down the decimal representation of
a googolplex

[

1

]

.

Properties

1.

A googol is approximately the

factorial

of

70

.
The only

prime factors

in a googol and a googolplex are

2

and

5

[

1

]

.

2.

Using Stirling’s

formula

we can approximate the factorial of a googol to obtain

10

100

!

≈

10

9.95

⋅

10

101

.

3.

A googoplex plus one is not a prime. One factor is 316912650057057350374175801344000001

[

5

]

.

History and etymology

The googol was created by the American mathematician Edward Kasner (1878-1955)

[

4

]

in

[

2

]

to illustrate the

difference

between an
unimaginably large number and

infinity

.
The name ’googol’ was supposedly coined by Kasner’s
nine-year-old nephew Milton Sirotta in 1938 when asked to give a name for a huge number.
The name googol was perhaps influenced by
the comic strip

character

Barney Google

[

1

,

3

]

.
The name of the search engine google was inspired by the number
googol

[

6

]

.

References

1

Wikipedia’s

http://en2.wikipedia.org/wiki/Googol

entry on googol.

2

Kasner, Edward&Newman, James Roy,

Mathematics and the Imagination

, (New York, NY, U
SA:
Simon and Schuster, 1967; Dover Pubns, April 2001; London: Penguin, 1940, ISBN 0486417034).

3

Douglas Harper’s

http://www.etymonline.com/g3etym.htm

Etymology online dictionary, Googol 12/2003.

4

Wikipedia’s

http://en2.wikipedia.org/wiki/Edward_Kasner

entry on Edward Kasner.

5

Robert Munafo,

http://www.mrob.com/pub/math/numbers-21.html

web page on large numbers.

6

Google corporate information:

http://www.google.com/corporate/history.html

Google Milestones.

Title

googol

Canonical

name

Googol

Date of creation

2013-03-22 12:25:31

Last modified on

2013-03-22 12:25:31

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

19

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 11-00

Related topic

Googolplex

Generated on Fri Feb 9 15:07:13 2018 by

LaTeXML
googolplex

googolplex

A

googolplex

is

10

10

100

; that is, 10 raised to the googol-th power. This can also be viewed as a one followed by a googol of zeros at its right. A googoplex is much larger than a googol. In fact, since a googolplex has
a googol number of zeros in its decimal representation, a googoplex has more digits than there are atoms in our universe. Thus, even if all matter in the universe were at one’s disposal, it would not be possible to write down the decimal

representation

of a googolplex.

Title

googolplex

Canonical name

Googolplex

Date of creation

2013-03-22 12:25:33

Last modified on

2013-03-22 12:25:33

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11-00

Related topic

Googol

Generated on Fri Feb 9 15:07:14 2018 by

LaTeXML