aarjaneiro/planetmath huggingface dataset:
Belgian number

Belgian number

108 is a Belgian-0 number because if we start a

sequence

with 0 with recurring

differences

of 1, 0 and 8 (digits of the number), 108 is in that sequence.The sequence is 1, 1, 9, 10, 10, 18, 19, 19, 27, 28, 28, 36, 37, 37, 45, 46, 46, 54, 55, 55, 63, 64, 64, 72, 73, 73, 81, 82, 82, 90, 91, 91, 99, 100, 100, 108.Similarly a number of the form abcd for example is called as a Belgian-n number if it belongs to the sequence starting with n and having reccuring differences of a, b, c, d …
The first few Belgian-0 numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 20, 21, 22, 24 … (Sequence A106039 of OEIS). The first few Belgian-1 numbers are 1, 10, 11, 13, 16, 17, 21, 23, 41, 43, 56, 58 … (Sequence A106439 of OEIS). The first few Belgian-2 numbers are 4, 10, 11, 13, 14, 20, 21, 22, 24, 25, 31, 32, 37, 40, 47 … (Sequence A106331 of OEIS). The first few Belgian-3 numbers are 3, 10, 11, 12, 14, 15, 21, 23, 30 … (Sequence A106596 of OEIS). The first few Belgian-4 numbers are 4, 10, 11, 13, 14, 20, 21, 22, 25 … (Sequence A106331 of OEIS).
All

Harshad numbers

are Belgian-0 numbers. For all k, the number of k-Belgian numbers are infinite. They have a

positive

density. Every

positive integer

is a Belgian-k number for some k.

Self-Belgian numbers are of two types. Self-Belgian number of type 1 is a number whose initial digit is ’a’ and is an a-Belgian number. For example 68 begins with 6 and is a 6-Belgian number. The self-Belgian numbers of type 1 are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 20, 22, 25, 26, 30, 31, 33, 34 … (Sequence A107062 of OEIS). There are infinitely meny Self-Belgian numbers of type 1.
Self

Belgian numbers

of type two are the Self-Belgian numbers of type 1 which fully show all their digits at the starting of the sequence. For example, 61 is a Self-Belgian number of type 2 as its sequence is 6, 12, 13, 19, 20, 26, 27, 33, 34, 40, 41, 47, 48, 54, 55 and 61, which begins with digits 6 and 1. The Self-Belgian numbers of type 2 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 61, 71, 918, 3612, 5101, 8161, 12481, 51011 … (Sequence A107070 of OEIS).

Title

Belgian number

Canonical

name

BelgianNumber

Date of creation

2013-03-22 19:20:45

Last modified on

2013-03-22 19:20:45

Owner

Kausthub (26471)

Last modified by

Kausthub (26471)

Numerical id

4

Author

Kausthub (26471)

Entry type

Definition

Classification

msc 11A25

Synonym

Eric number

Generated on Fri Feb 9 15:12:45 2018 by

LaTeXML
Bombieri-Vinogradov theorem

Bombieri-Vinogradov theorem

The

Bombieri-Vinogradov theorem

, sometimes called

Bombieri’s theorem

, states that for a positive real number

A

, if

x

1

2

⁢

log

-

A

⁡

x

≤

Q

≤

x

1

2

then

∑

q

≤

Q

max

y

≤

x

⁡

max

1

≤

a

≤

q

(

a

,

q

)

=

1

⁡

|

ψ

⁢

(

x

;

q

,

a

)

-

x

ϕ

⁢

(

q

)

|

=

O

⁢

(

x

1

2

⁢

Q

⁢

(

log

⁡

x

)

5

)

,

where

ϕ

⁢

(

q

)

is Euler’s

totient function

and

ψ

⁢

(

x

;

q

,

a

)

=

∑

n

≤

x

n

≡

a

mod

q

Λ

⁢

(

n

)

,

where

Λ

⁢

(

n

)

is the

Mangoldt function

.

Title

Bombieri-Vinogradov theorem

Canonical name

BombieriVinogradovTheorem

Date of creation

2013-03-22 16:25:36

Last modified on

2013-03-22 16:25:36

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

4

Author

Mravinci (12996)

Entry type

Theorem

Classification

msc 11A25

Synonym

Bombieri’s theorem

Generated on Fri Feb 9 15:12:47 2018 by

LaTeXML
Brocard’s problem

Brocard’s problem

Brocard’s problem

, first posed by Henri Brocard in 1876, asks for

factorials

that are one less than a square, that is, solutions to the equation

n

!

+

1

=

m

2

. Only three solutions are known:

4

!

+

1

=

5

2

,

5

!

+

1

=

11

2

and

7

!

+

1

=

71

2

.

Srinivasa Ramanujan

also pondered the problem, in 1913. Erdős believed that there are no other solutions, and no more have been found for

n

up to

10

9

.

References

1

P. Erdős, & R. OblÃÂ¡th, “Über diophantische Gleichungen der Form

n

!

=

x

p

±

y

p

und

n

!

±

m

!

=

x

p

”

Acta Szeged.

8

(1937): 241 - 255

Title

Brocard’s problem

Canonical

name

BrocardsProblem

Date of creation

2013-03-22 18:09:59

Last modified on

2013-03-22 18:09:59

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Generated on Fri Feb 9 15:12:49 2018 by

LaTeXML
Carmichael function

Carmichael function

The value of the

Carmichael function

(or

universal

exponent

function

or

reduced totient function

)

ψ

⁢

(

n

)

(or

λ

⁢

(

n

)

) for a given

positive integer

n

is the smallest exponent

m

such that for any

k

coprime

to

n

the

congruence

k

m

≡

1

mod

n

is always true.

When

n

is a prime or a square of a prime, the equality

ψ

⁢

(

n

)

=

ϕ

⁢

(

n

)

holds (where

ϕ

⁢

(

n

)

is Euler’s

totient function

).

For powers of 2 greater than 4,

ψ

⁢

(

2

a

)

=

ϕ

⁢

(

2

a

)

2

(with

a

>

2

).

For all other positive integers the value of Carmichael’s function is the least common

multiple

of all the dividing primes raised to the appropriate powers (e.g., to calculate

ψ

⁢

(

504

)

we’d reckon

ψ

⁢

(

2

3

)

,

ψ

⁢

(

3

2

)

and

ψ

⁢

(

7

)

and find the LCM of these).

Sequence

A002322 in Sloane’s OEIS gives values of

ψ

⁢

(

n

)

for

0

<

n

<

82

.

So, for example,

ψ

⁢

(

16

)

=

4

. This means that any

odd number

raised to the

fourth power

is one more than a multiple of 16. If we take a few small odd numbers in order and raise them to the fourth power, we get the sequence 1, 81, 625, 2401, 6561, 14641, 28561, 50625, 83521, 130321, 194481. Subtracting 1 from each of these and dividing by 16 we get the

integers

0, 5, 39, 150, 410, 915, 1785, 3164, 5220, 8145, 12155.

Of course from Fermat’s little

theorem

we can deduce that

ϕ

⁢

(

n

)

will give us an exponent to which we can raise any number coprime to

n

and get a number satisfying the congruence.

ψ

⁢

(

n

)

often gives us a smaller exponent than

ϕ

⁢

(

n

)

for

composite

n

that are not squares of primes. Among the first thousand positive integers, this is true 86% of the time. Sequence A104194 gives

ϕ

⁢

(

n

)

-

ψ

⁢

(

n

)

for

0

<

n

<

91

; it has many

instances

of 0.

In Sloane and Plouffe’s book

The Encyclopedia of

Integer Sequences

the authors use the Greek letter

ψ

for this function, the OEIS follows this custom but acknowledges the widespread use of

λ

. In Mathematica, the function is a built-in function,

CarmichaelLambda[n]

, so naturally Mathworld also uses

λ

, and so does Wikipedia.

References

1

G. P. Lowecke,

The Lore of

Prime Numbers

. New York: Vantage Press (1982): 81 - 82

2

H. Griffin,

Elementary Theory of Numbers

. New York: McGraw-Hill (1954): 50

3

N. Sloane & S. Plouffe

The Encyclopedia of Integer Sequences

New York: Academic Press (1995): N0110

4

I. Vardi,

Computational Recreations in Mathematica

. Redwood City: Addison-Wesley (1991): 226

Title

Carmichael function

Canonical name

CarmichaelFunction

Date of creation

2013-03-22 17:13:47

Last modified on

2013-03-22 17:13:47

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Synonym

least universal exponent function

Synonym

reduced totient function

Generated on Fri Feb 9 15:12:51 2018 by

LaTeXML
character of a finite group

character of a finite group

Definition

.

Let

G

be a

finite group

, and let

K

be a field. A

character

from

G

to

K

is a

group homomorphism

χ

:

G

→

K

×

, where

K

×

is the

multiplicative group

K

∖

{

0

K

}

.

Example

:
The

Dirichlet characters

are characters from

ℤ

/

m

⁢

ℤ

to

ℂ

.

Title

character of a finite group

Canonical name

CharacterOfAFiniteGroup

Date of creation

2013-03-22 14:10:27

Last modified on

2013-03-22 14:10:27

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 11A25

Synonym

character

Related topic

DirichletCharacter

Generated on Fri Feb 9 15:12:53 2018 by

LaTeXML
completely multiplicative functions whose convolution inverses are completely multiplicative

completely multiplicative functions whose convolution inverses are completely multiplicative

Corollary 1

.

The only

completely multiplicative function

whose convolution inverse is also completely multiplicative is

ε

, the convolution identity function.

Proof.

Let

f

be a completely multiplicative function whose convolution inverse is completely multiplicative. By this entry (

http://planetmath.org/FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction

),

f

⁢

μ

is the convolution inverse of

f

, where

μ

denotes the Möbius

function

. Thus,

f

⁢

μ

is completely multiplicative.

Let

p

be any prime. Then

(

f

⁢

(

p

)

)

2

=

(

f

⁢

(

p

)

)

2

⁢

(

-

1

)

2

=

(

f

⁢

(

p

)

)

2

⁢

(

μ

⁢

(

p

)

)

2

=

(

f

⁢

(

p

)

⁢

μ

⁢

(

p

)

)

2

=

f

⁢

(

p

2

)

⁢

μ

⁢

(

p

2

)

=

f

⁢

(

p

2

)

⋅

0

=

0

.

Thus,

f

⁢

(

p

)

=

0

for every prime

p

. Since

f

is completely multiplicative,

f

⁢

(

n

)

=

{

1

if

⁢

n

=

1

0

if

⁢

n

≠

1

.

Hence,

f

=

ε

.
∎

Title

completely multiplicative functions whose convolution inverses are completely multiplicative

Canonical

name

CompletelyMultiplicativeFunctionsWhoseConvolutionInversesAreCompletelyMultiplicative

Date of creation

2013-03-22 16:55:12

Last modified on

2013-03-22 16:55:12

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

4

Author

Wkbj79 (1863)

Entry type

Corollary

Classification

msc 11A25

Generated on Fri Feb 9 15:12:55 2018 by

LaTeXML
composition of multiplicative functions

composition of multiplicative functions

Theorem

.

If

f

is a

completely multiplicative function

and

g

is a

multiplicative function

, then

f

∘

g

is a multiplicative function.

Proof.

First note that

(

f

∘

g

)

⁢

(

1

)

=

f

⁢

(

g

⁢

(

1

)

)

=

f

⁢

(

1

)

=

1

since both

f

and

g

are multiplicative.

Let

a

and

b

be relatively prime

positive integers

. Then

(

f

∘

g

)

⁢

(

a

⁢

b

)

=

f

⁢

(

g

⁢

(

a

⁢

b

)

)

=

f

⁢

(

g

⁢

(

a

)

⋅

g

⁢

(

b

)

)

since

g

is multiplicative

=

f

⁢

(

g

⁢

(

a

)

)

⁢

f

⁢

(

g

⁢

(

b

)

)

since

f

is completely multiplicative

=

(

f

∘

g

)

⁢

(

a

)

⁢

(

f

∘

g

)

⁢

(

b

)

.

∎

Note that the

assumption

that

f

is

completely

multiplicative (as opposed to merely multiplicative) is essential in proving that

f

∘

g

is multiplicative. For

instance

,

τ

∘

τ

, where

τ

denotes the

divisor function

, is not multiplicative:

(

τ

∘

τ

)

⁢

(

2

⋅

3

)

=

(

τ

∘

τ

)

⁢

(

6

)

=

τ

⁢

(

τ

⁢

(

6

)

)

=

τ

⁢

(

4

)

=

3

(

τ

∘

τ

)

⁢

(

2

)

⋅

(

τ

∘

τ

)

⁢

(

3

)

=

τ

⁢

(

τ

⁢

(

2

)

)

⋅

τ

⁢

(

τ

⁢

(

3

)

)

=

τ

⁢

(

2

)

⋅

τ

⁢

(

2

)

=

2

⋅

2

=

4

Title

composition of multiplicative functions

Canonical name

CompositionOfMultiplicativeFunctions

Date of creation

2013-03-22 16:09:50

Last modified on

2013-03-22 16:09:50

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11A25

Generated on Fri Feb 9 15:12:58 2018 by

LaTeXML
convolution inverses for arithmetic functions

convolution inverses for arithmetic functions

Theorem

.

An

arithmetic function

f

has a

convolution inverse

if and only if

f

⁢

(

1

)

≠

0

.

Proof.

If

f

has a convolution inverse

g

, then

f

*

g

=

ε

, where

ε

denotes the convolution identity function. Thus,

1

=

ε

⁢

(

1

)

=

(

f

*

g

)

⁢

(

1

)

=

f

⁢

(

1

)

⁢

g

⁢

(

1

)

, and it follows that

f

⁢

(

1

)

≠

0

.

Conversely

, if

f

⁢

(

1

)

≠

0

, then an arithmetic function

g

must be constructed such that

(

f

*

g

)

⁢

(

n

)

=

ε

⁢

(

n

)

for all

n

∈

ℕ

. This will be done by

induction

on

n

.

Since

f

⁢

(

1

)

≠

0

, we have that

1

f

⁢

(

1

)

∈

ℂ

. Define

g

⁢

(

1

)

=

1

f

⁢

(

1

)

.

Now let

k

∈

ℕ

with

k

>

1

and

g

⁢

(

1

)

,

…

,

g

⁢

(

k

-

1

)

be such that

(

f

*

g

)

⁢

(

n

)

=

ε

⁢

(

n

)

for all

n

∈

ℕ

with

n

<

k

.

Define

g

⁢

(

k

)

=

-

1

f

⁢

(

1

)

⁢

∑

d

|

k

and

d

<

k

f

⁢

(

k

d

)

⁢

g

⁢

(

d

)

.

Then

(

f

*

g

)

⁢

(

k

)

=

∑

d

|

k

f

⁢

(

k

d

)

⁢

g

⁢

(

d

)

=

f

⁢

(

1

)

⁢

g

⁢

(

k

)

+

∑

d

|

k

and

d

<

k

f

⁢

(

k

d

)

⁢

g

⁢

(

d

)

=

f

⁢

(

1

)

⁢

(

-

1

f

⁢

(

1

)

⁢

∑

d

|

k

and

d

<

k

f

⁢

(

k

d

)

⁢

g

⁢

(

d

)

)

+

∑

d

|

k

and

d

<

k

f

⁢

(

k

d

)

⁢

g

⁢

(

d

)

=

0

=

ε

⁢

(

k

)

.

∎

In the entry titled arithmetic functions form a ring, it is proven that convolution is

associative

and

commutative

. Thus,

G

=

{

f

:

ℕ

→

ℂ

|

f

(

1

)

≠

0

}

is an

abelian group

under convolution. The set of all

multiplicative functions

is a

subgroup

of

G

.

Title

convolution inverses for arithmetic functions

Canonical name

ConvolutionInversesForArithmeticFunctions

Date of creation

2013-03-22 15:58:32

Last modified on

2013-03-22 15:58:32

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

27

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11A25

Related topic

ArithmeticFunction

Related topic

MultiplicativeFunction

Related topic

ArithmeticFunctionsFormARing

Related topic

ElementaryResultsAboutMultiplicativeFunctionsAndConvolution

Generated on Fri Feb 9 15:13:01 2018 by

LaTeXML
criterion for a multiplicative function to be completely multiplicative

criterion for a multiplicative function to be completely multiplicative

Theorem

.

Let

f

be a

multiplicative function

with convolution inverse

g

. Then

f

is completely multiplicative if and only if

g

⁢

(

p

k

)

=

0

for all primes

p

and for all

k

∈

N

with

k

>

1

.

Proof.

Note first that, since

f

⁢

(

1

)

=

1

and

f

*

g

=

ε

, where

ε

denotes the convolution identity function, then

g

⁢

(

1

)

=

1

. Let

p

be any prime. Then

0

=

ε

⁢

(

p

)

=

(

f

*

g

)

⁢

(

p

)

=

f

⁢

(

1

)

⁢

g

⁢

(

p

)

+

f

⁢

(

p

)

⁢

g

⁢

(

1

)

=

g

⁢

(

p

)

+

f

⁢

(

p

)

.

Thus,

g

⁢

(

p

)

=

-

f

⁢

(

p

)

.

Assume that

f

is completely multiplicative. The statement about

g

will be proven by

induction

on

k

. Note that:

0

=

ε

⁢

(

p

2

)

=

(

f

*

g

)

⁢

(

p

2

)

=

f

⁢

(

1

)

⁢

g

⁢

(

p

2

)

+

f

⁢

(

p

)

⁢

g

⁢

(

p

)

+

f

⁢

(

p

2

)

⁢

g

⁢

(

1

)

=

g

⁢

(

p

2

)

+

f

⁢

(

p

)

⁢

(

-

f

⁢

(

p

)

)

+

(

f

⁢

(

p

)

)

2

=

g

⁢

(

p

2

)

Let

m

∈

ℕ

with

m

>

2

such that, for all

k

∈

ℕ

with

1

<

k

<

m

,

g

⁢

(

p

k

)

=

0

. Then:

0

=

ε

⁢

(

p

m

)

=

(

f

*

g

)

⁢

(

p

m

)

=

f

⁢

(

1

)

⁢

g

⁢

(

p

m

)

+

f

⁢

(

p

m

-

1

)

⁢

g

⁢

(

p

)

+

f

⁢

(

p

m

)

⁢

g

⁢

(

1

)

=

g

⁢

(

p

m

)

+

(

f

⁢

(

p

)

)

m

-

1

⁢

(

-

f

⁢

(

p

)

)

+

(

f

⁢

(

p

)

)

m

=

g

⁢

(

p

m

)

Conversely

, assume that

g

⁢

(

p

k

)

=

0

for all

k

∈

ℕ

with

k

>

1

. The statement

f

⁢

(

p

k

)

=

(

f

⁢

(

p

)

)

k

will be proven by induction on

k

. The statement is obvious for

k

=

1

. Let

m

∈

ℕ

such that

f

⁢

(

p

m

-

1

)

=

(

f

⁢

(

p

)

)

m

-

1

. Then:

0

=

ε

⁢

(

p

m

)

=

(

f

*

g

)

⁢

(

p

m

)

=

f

⁢

(

p

m

-

1

)

⁢

g

⁢

(

p

)

+

f

⁢

(

p

m

)

⁢

g

⁢

(

1

)

=

(

f

⁢

(

p

)

)

m

-

1

⁢

(

-

f

⁢

(

p

)

)

+

f

⁢

(

p

m

)

=

-

(

f

⁢

(

p

)

)

m

+

f

⁢

(

p

m

)

Thus,

f

⁢

(

p

m

)

=

(

f

⁢

(

p

)

)

m

. It follows that

f

is completely multiplicative.
∎

Title

criterion for a multiplicative function to be completely multiplicative

Canonical name

CriterionForAMultiplicativeFunctionToBeCompletelyMultiplicative

Date of creation

2013-03-22 15:58:44

Last modified on

2013-03-22 15:58:44

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11A25

Related topic

FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction

Generated on Fri Feb 9 15:13:04 2018 by

LaTeXML
Dirichlet character

Dirichlet character

A

Dirichlet character

modulo

m

is a

group homomorphism

from

(

ℤ

m

⁢

ℤ

)

*

to

ℂ

*

. Dirichlet characters are usually denoted by the Greek letter

χ

. The

function

γ

⁢

(

n

)

=

{

χ

⁢

(

n

mod

m

)

,

if

⁢

gcd

⁡

(

n

,

m

)

=

1

,

0

,

if

⁢

gcd

⁡

(

n

,

m

)

>

1

.

is also referred to as a Dirichlet character.
The Dirichlet characters modulo

m

form a group if one defines

χ

⁢

χ

′

to be the function which takes

a

∈

(

ℤ

m

⁢

ℤ

)

*

to

χ

⁢

(

a

)

⁢

χ

′

⁢

(

a

)

. It turns out that this resulting group is

isomorphic

to

(

ℤ

m

⁢

ℤ

)

*

. The trivial character is given by

χ

⁢

(

a

)

=

1

for all

a

∈

(

ℤ

m

⁢

ℤ

)

*

, and it acts as the

identity element

for the group.
A

character

χ

modulo

m

is said to be

induced

by a character

χ

′

modulo

m

′

if

m

′

∣

m

and

χ

⁢

(

n

)

=

χ

′

⁢

(

n

mod

m

′

)

. A character which is not induced by any other character is called

primitive

.
If

χ

is non-primitive, the

gcd

of all such

m

′

is called the

conductor

of

χ

.

Examples:

•

Legendre symbol

(

n

p

)

is a Dirichlet character modulo

p

for any

odd prime

p

. More generally,

Jacobi symbol

(

n

m

)

is a Dirichlet character modulo

m

.

•

The character modulo

4

given by

χ

⁢

(

1

)

=

1

and

χ

⁢

(

3

)

=

-

1

is a

primitive character

modulo

4

. The only other character modulo

4

is the trivial character.

Title

Dirichlet character

Canonical name

DirichletCharacter

Date of creation

2013-03-22 13:22:31

Last modified on

2013-03-22 13:22:31

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

10

Author

bbukh (348)

Entry type

Definition

Classification

msc 11A25

Related topic

CharacterOfAFiniteGroup

Defines

trivial character

Defines

primitive character

Defines

conductor

Defines

induced character

Generated on Fri Feb 9 15:13:06 2018 by

LaTeXML
divisor function is multiplicative, the

divisor function is multiplicative, the

Theorem

.

The

divisor function

(

http://planetmath.org/TauFunction

) is

multiplicative

.

Proof.

Let

t

=

m

⁢

n

with

m

,

n

coprime

.
Applying the

fundamental theorem of arithmetic

, we can write

m

=

p

1

a

1

⁢

p

2

a

2

⁢

⋯

⁢

p

r

a

r

,

n

=

q

1

b

1

⁢

q

2

b

2

⁢

⋯

⁢

q

s

b

s

,

where each

p

j

and

q

i

are prime.
Moreover, since

m

and

n

are coprime, we conclude that

t

=

p

1

a

1

⁢

p

2

a

2

⁢

⋯

⁢

p

r

a

r

⁢

q

1

b

1

⁢

q

2

b

2

⁢

⋯

⁢

q

s

b

s

.

Now, each

divisor

of

t

is of the form

t

=

p

1

k

1

⁢

p

2

k

2

⁢

⋯

⁢

p

r

k

r

⁢

q

1

h

1

⁢

q

2

h

2

⁢

⋯

⁢

q

s

h

s

.

with

0

≤

k

j

≤

a

j

and

0

≤

h

i

≤

b

i

,
and for each such divisor we get a divisor of

m

and a divisor of

n

,
given respectively by

u

=

p

1

k

1

⁢

p

2

k

2

⁢

⋯

⁢

p

r

k

r

,

v

=

q

1

h

1

⁢

q

2

h

2

⁢

⋯

⁢

q

s

h

s

.

Now, each respective divisor of

m

,

n

is of the form above,
and for each such pair their product is also a divisor of

t

.
Therefore we get a bijection between the set of

positive

divisors of

t

and the set of pairs of divisors of

m

,

n

respectively.
Such bijection implies that the cardinalities of both sets are the same,
and thus

d

⁢

(

m

⁢

n

)

=

d

⁢

(

m

)

⁢

d

⁢

(

n

)

.

Title

divisor function is multiplicative, the

Canonical name

DivisorFunctionIsMultiplicativeThe

Date of creation

2013-03-22 15:03:47

Last modified on

2013-03-22 15:03:47

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Theorem

Classification

msc 11A25

Generated on Fri Feb 9 15:13:08 2018 by

LaTeXML
divisor sum of an arithmetic function

divisor sum of an arithmetic function

Given an

arithmetic function

f

:

ℤ

+

→

ℂ

, the

divisor sum

F

of

f

is defined by

F

⁢

(

n

)

=

∑

d

∣

n

f

⁢

(

d

)

⁢

,

(1)

where the

summation

runs over all

positive

divisors

d

of

n

.

Title

divisor sum of an arithmetic function

Canonical

name

DivisorSumOfAnArithmeticFunction

Date of creation

2013-03-22 17:07:31

Last modified on

2013-03-22 17:07:31

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

10

Author

azdbacks4234 (14155)

Entry type

Definition

Classification

msc 11A25

Related topic

Divisibility

Related topic

ArithmeticFunction

Related topic

MultiplicativeFunction

Defines

divisor sum

Generated on Fri Feb 9 15:13:10 2018 by

LaTeXML
Euler phi at a product

Euler phi at a product

If the

positive

greatest common divisor

of the

integers

a

and

b

is

d

, then

φ

⁢

(

a

⁢

b

)

=

φ

⁢

(

a

)

⁢

φ

⁢

(

b

)

⁢

d

φ

⁢

(

d

)

.

Proof.

Using the positive

prime factors

p

, the right hand side of the asserted equation is

d

⋅

a

⁢

∏

p

∣

a

p

-

1

p

⋅

b

⁢

∏

p

∣

b

p

-

1

p

d

⁢

∏

p

∣

a

,

p

∣

b

p

-

1

p

=

a

⁢

b

⁢

∏

p

∣

a

,

p

∤

b

p

-

1

p

⋅

∏

p

∣

a

,

p

∣

b

p

-

1

p

⋅

∏

p

∣

b

,

p

∤

a

p

-

1

p

⋅

∏

p

∣

b

,

p

∣

a

p

-

1

p

∏

p

∣

a

,

p

∣

b

p

-

1

p

=

a

⁢

b

⁢

∏

p

⁢

∣

a

∨

p

∣

⁢

b

p

-

1

p

=

a

⁢

b

⁢

∏

p

∣

a

b

p

-

1

p

=

φ

⁢

(

a

⁢

b

)

,

Q.E.D.

Title

Euler phi at a

product

Canonical name

EulerPhiAtAProduct

Date of creation

2014-02-18 14:02:24

Last modified on

2014-02-18 14:02:24

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A25

Classification

msc 11-00

Related topic

EulerPhifunction

Related topic

DivisibilityByPrimeNumber

Generated on Fri Feb 9 15:13:17 2018 by

LaTeXML