aarjaneiro/planetmath huggingface dataset:
Euler four-square identity

Euler four-square identity

The

Euler four-square identity

simply states that

(

x

1

2

+

x

2

2

+

x

3

2

+

x

4

2

)

⁢

(

y

1

2

+

y

2

2

+

y

3

2

+

y

4

2

)

=

(

x

1

⁢

y

1

+

x

2

⁢

y

2

+

x

3

⁢

y

3

+

x

4

⁢

y

4

)

2

+

(

x

1

⁢

y

2

-

x

2

⁢

y

1

+

x

3

⁢

y

4

-

x

4

⁢

y

3

)

2

+

(

x

1

⁢

y

3

-

x

3

⁢

y

1

+

x

4

⁢

y

2

-

x

2

⁢

y

4

)

2

+

(

x

1

⁢

y

4

-

x

4

⁢

y

1

+

x

2

⁢

y

3

-

x

3

⁢

y

2

)

2

It may be derived from the property of quaternions that the norm of the product is equal to the product of the norms.

Title

Euler four-square identity

Canonical name

EulerFoursquareIdentity

Date of creation

2013-03-22 12:35:20

Last modified on

2013-03-22 12:35:20

Owner

vitriol (148)

Last modified by

vitriol (148)

Numerical id

6

Author

vitriol (148)

Entry type

Theorem

Classification

msc 11N32

Related topic

Quaternions

Related topic

MultiplicativityOfSumsOfSquares

Related topic

LagrangesFourSquareTheorem

Related topic

SumsOfTwoSquares

Generated on Fri Feb 9 15:39:18 2018 by

LaTeXML
Schinzel’s Hypothesis H

Schinzel’s Hypothesis H

Let a set of

irreducible polynomials

P

1

,

P

2

,

P

3

,

…

,

P

k

with

integer

coefficients

have the

property

that for any prime

p

there exists some

n

such that

P

1

⁢

(

n

)

⁢

P

2

⁢

(

n

)

⁢

…

⁢

P

k

⁢

(

n

)

≢

0

⁢

(

m

⁢

o

⁢

d

⁢

p

)

. Schinzel’s

Hypothesis

H that there are infinitely many values of

n

for which

P

1

⁢

(

n

)

,

P

2

⁢

(

n

)

,

…

,

and

P

k

⁢

(

n

)

are all

prime numbers

.

The 1st condition is necessary since if

P

i

is reducible then

P

i

⁢

(

n

)

cannot be prime except in the finite number of cases where all but one of its factors are equal to 1 or -1. The second condition is necessary as otherwise there will always be at least 1 of the

P

i

⁢

(

n

)

divisible

by

p

; and thus not all of the

P

i

⁢

(

n

)

are prime except in the finite number of cases where one of the

P

i

⁢

(

n

)

is equal to

p

.

It includes several other

conjectures

, such as the

twin prime conjecture

.

Title

Schinzel’s Hypothesis H

Canonical

name

SchinzelsHypothesisH

Date of creation

2013-03-22 15:11:43

Last modified on

2013-03-22 15:11:43

Owner

jtolliver (9126)

Last modified by

jtolliver (9126)

Numerical id

5

Author

jtolliver (9126)

Entry type

Conjecture

Classification

msc 11N32

Generated on Fri Feb 9 15:39:20 2018 by

LaTeXML
wheel factorization

wheel factorization

Wheel factorization

is a method for screening out

composite numbers

by arranging numbers around a circle and then striking out those numbers that are obviously composite as indicated by where they fall around the circle.

Wheel factorization is performed thus:

1.

Use another method to find the first

i

primes. Multiply the first

i

primes, obtaining the

primorial

p

i

⁢

#

.

2.

Around a circle, write the numbers 1 to

p

i

⁢

#

.

3.

On a circle larger than the innermost circle, write the

p

i

⁢

#

+

1

to

2

⁢

p

i

⁢

#

, lining up 1 with

p

i

⁢

#

+

1

and

2

⁢

p

i

⁢

#

.

4.

Make a few more circles, as desired, but always aligning the numbers according to their

residues

modulo

p

i

⁢

#

.

5.

Circle the first

i

primes, and strike out those numbers that are lined up with them in the outer circles.

6.

For each prime

p

j

from 2 to

p

i

, strike out its

multiples

from

p

j

2

to

p

i

⁢

#

, as well as the numbers on the outer circles with the same residue modulo the primorial.

The name is somewhat misleading, as wheel factorization does not completely factor any number, but merely identifies (in the case of the struck-out numbers) one of their

prime factors

. All wheel factorization does is identify which numbers are

coprime

to the primorial (the numbers that have not been struck out). Of course the

prime numbers

greater than the primorial will not be struck out, since they must of course by definition be coprime to the the primorial. But as one does more and more circles, more and more composites will pass undetected by this method, as the likelihood increases that their least prime factor is a primer greater than the primorial. Even the innermost circle will always have at least one composite number avoid being struck out, whenever

i

>

3

, specifically

p

i

+

1

2

. And there is no practical reason why this algorithm needs to be performed on a circle, other than that it is more visually interesting to humans than a rectangular table. Indeed, with the

sieve of Eratosthenes

, if one has chosen a row width that is a multiple of a primorial, one can strike out entire columns at once.

Title

wheel factorization

Canonical name

WheelFactorization

Date of creation

2013-03-22 18:22:58

Last modified on

2013-03-22 18:22:58

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N35

Generated on Fri Feb 9 15:39:25 2018 by

LaTeXML
Brun’s constant

Brun’s constant

Brun’s constant

is the sum of the

reciprocals

of all twin primes

B

=

∑

p

p

+

2

⁢

is prime

(

1

p

+

1

p

+

2

)

≈

1.9216058

.

Viggo Brun proved that the constant exists by using a new sieving method, which later became known as Brun’s sieve (

http://planetmath.org/BrunsPureSieve

).

Title

Brun’s constant

Canonical

name

BrunsConstant

Date of creation

2013-03-22 13:20:01

Last modified on

2013-03-22 13:20:01

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

8

Author

bbukh (348)

Entry type

Definition

Classification

msc 11N36

Classification

msc 11N05

Related topic

BrunsPureSieve

Generated on Fri Feb 9 15:39:27 2018 by

LaTeXML
analytic number theory

analytic number theory

Analytic number theory

uses the machinery of

analysis

to tackle questions related to

integers

and transcendence. One of its most famous achievements is the proof of the

prime number theorem

.

One concept that is important in analytic number theory is

asymptotic estimates

. Tools that are used to obtain asymptotic estimates for sums include the

Euler-Maclaurin summation formula

(

http://planetmath.org/EulerMaclaurinSummationFormula

), Abel’s lemma (

summation by parts

), the

convolution method

, and the

Dirichlet hyperbola method

. Asymptotic estimates are important for determining

asymptotic densities

of certain subsets of the natural numbers.

Another one of Dirichlet’s contributions to analytic number theory is the

Dirichlet series

. As an example, the Dirichlet series of a

Dirichlet character

is a

Dirichlet L-series

. A tool that is helpful for studying any Dirichlet series is the

Euler product

. The most famous Dirichlet series is the

Riemann zeta function

, which is the Dirichlet series of the

completely multiplicative function

1

. This leads up to what is possibly the most important unsolved problem in analytic number theory: the

Riemann hypothesis

. This that all nontrivial zeros of the Riemann zeta function have

real part

equal to

1

2

. Its to

prime numbers

is made clearer by the

Euler product formula

.

Title

analytic number theory

Canonical

name

AnalyticNumberTheory

Date of creation

2013-03-22 15:59:55

Last modified on

2013-03-22 15:59:55

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 11N37

Classification

msc 11M06

Classification

msc 11N05

Classification

msc 11-01

Generated on Fri Feb 9 15:39:35 2018 by

LaTeXML
asymptotic estimate

asymptotic estimate

\PMlinkescapephrase

characteristic function

An

asymptotic estimate

is an that involves the use of

O

,

o

, or

∼

. These are all defined in the entry

Landau notation

. Examples of

asymptotic

are:

∑

n

≤

x

μ

2

⁢

(

n

)

=

6

π

2

⁢

x

+

O

⁢

(

x

)

(see

convolution method

for more details)

π

⁢

(

x

)

∼

x

log

⁡

x

(see

prime number theorem

for more details)

Unless otherwise specified, asymptotic are typically valid for

x

→

∞

. An example of an asymptotic that is different from those above in this aspect is

cos

⁡

x

=

1

-

x

2

2

+

O

⁢

(

x

4

)

⁢

for

⁢

|

x

|

<

1

.

Note that the above would be undesirable for

x

→

∞

, as the would be larger than the . Such is not the case for

|

x

|

<

1

, though.

Tools that are useful for obtaining asymptotic include:

•

the

Euler-Maclaurin summation formula

•

Abel’s lemma

•

the convolution method (

http://planetmath.org/ConvolutionMethod

)

•

the

Dirichlet hyperbola method

If

A

⊆

ℕ

, then an asymptotic for

∑

n

≤

x

χ

A

⁢

(

x

)

, where

χ

A

denotes the characteristic function (

http://planetmath.org/CharacteristicFunction

) of

A

, enables one to determine the

asymptotic density

of

A

using the

lim

x

→

∞

⁡

1

x

⁢

∑

n

≤

x

χ

A

⁢

(

x

)

provided the limit exists. The upper asymptotic density of

A

and the lower asymptotic density of

A

can be computed in a manner using

lim sup

and

lim inf

, respectively. (See asymptotic density (

http://planetmath.org/AsymptoticDensity

) for more details.)

For example,

μ

2

is the characteristic function of the

squarefree

natural numbers

. Using the asymptotic above yields the asymptotic density of the squarefree natural numbers:

lim

x

→

∞

⁡

1

x

⁢

∑

n

≤

x

μ

2

⁢

(

n

)

=

lim

x

→

∞

⁡

1

x

⁢

(

6

π

2

⁢

x

+

O

⁢

(

x

)

)

=

lim

x

→

∞

⁡

6

π

2

+

O

⁢

(

x

x

)

=

6

π

2

Title

asymptotic estimate

Canonical

name

AsymptoticEstimate

Date of creation

2013-03-22 16:00:01

Last modified on

2013-03-22 16:00:01

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11N37

Related topic

AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions

Related topic

DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2

Related topic

DisplaystyleXlog2xOleftsum_nLeX2OmeganRight

Related topic

DisplaystyleSum_nLeXYomeganO_yxlogXy1ForYGe0

Generated on Fri Feb 9 15:39:37 2018 by

LaTeXML
convolution method

convolution method

Let

f

,

g

, and

h

be

multiplicative functions

such that

f

=

g

*

h

, where

*

denotes the

convolution

(

http://planetmath.org/MultiplicativeFunction

) of

g

and

h

. The

convolution method

is a way to

∑

n

≤

x

f

⁢

(

n

)

by using the fact that

f

=

g

*

h

:

∑

n

≤

x

f

⁢

(

n

)

=

∑

n

≤

x

∑

d

|

n

g

⁢

(

d

)

⁢

h

⁢

(

n

d

)

=

∑

d

≤

x

∑

m

≤

x

d

g

⁢

(

d

)

⁢

h

⁢

(

m

)

=

∑

d

≤

x

g

⁢

(

d

)

⁢

∑

m

≤

x

d

h

⁢

(

m

)

This method for calculating

∑

n

≤

x

f

⁢

(

n

)

is advantageous when the sums in of

g

and

h

are easier to handle.

As an example, the sum

∑

n

≤

x

μ

2

⁢

(

n

)

will be calculated using the convolution method.

Since

μ

2

=

μ

2

*

ε

=

μ

2

*

(

μ

*

1

)

=

(

μ

2

*

μ

)

*

1

, the

functions

μ

2

*

μ

and

1

can be used.

To use the convolution method, a nice way to

(

μ

2

*

μ

)

⁢

(

n

)

needs to be found. Note that

μ

2

*

μ

is multiplicative (

http://planetmath.org/MultiplicativeFunction

), so it only needs to be evaluated at prime powers.

Let

k

∈

ℕ

. Then

(

μ

2

*

μ

)

⁢

(

p

k

)

=

∑

j

=

0

k

μ

2

⁢

(

p

j

)

⁢

μ

⁢

(

p

k

-

j

)

=

μ

⁢

(

p

k

)

+

μ

⁢

(

p

k

-

1

)

=

{

0

if

⁢

k

≠

2

-

1

if

⁢

k

=

2

.

Since

μ

2

*

μ

is multiplicative (

http://planetmath.org/MultiplicativeFunction

), then

(

μ

2

*

μ

)

⁢

(

n

)

=

{

0

if

⁢

n

⁢

is not a square

μ

⁢

(

n

)

if

⁢

n

⁢

is a square.

The convolution method yields:

∑

n

≤

x

μ

2

⁢

(

n

)

=

∑

d

≤

x

(

μ

2

*

μ

)

⁢

(

d

)

⁢

∑

m

≤

x

d

1

=

∑

d

≤

x

(

μ

2

*

μ

)

⁢

(

d

)

⁢

(

x

d

+

O

⁢

(

1

)

)

=

∑

a

≤

x

μ

⁢

(

a

)

⁢

(

x

a

2

+

O

⁢

(

1

)

)

=

x

⁢

∑

a

≤

x

μ

⁢

(

a

)

a

2

+

O

⁢

(

∑

a

≤

x

|

μ

⁢

(

a

)

|

)

=

x

⁢

(

∑

a

∈

ℕ

μ

⁢

(

a

)

a

2

-

∑

a

>

x

μ

⁢

(

a

)

a

2

)

+

O

⁢

(

∑

a

≤

x

1

)

=

x

⁢

∑

a

∈

ℕ

μ

⁢

(

a

)

a

2

+

O

⁢

(

x

⁢

∑

a

>

x

|

μ

⁢

(

a

)

a

2

|

)

+

O

⁢

(

x

)

=

6

π

2

⁢

x

+

O

⁢

(

x

⁢

∑

a

>

x

1

a

2

)

+

O

⁢

(

x

)

=

6

π

2

⁢

x

+

O

⁢

(

x

⁢

(

1

x

)

)

+

O

⁢

(

x

)

=

6

π

2

⁢

x

+

O

⁢

(

x

)

Title

convolution method

Canonical name

ConvolutionMethod

Date of creation

2013-03-22 15:58:24

Last modified on

2013-03-22 15:58:24

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11N37

Related topic

DirichletHyperbolaMethod

Related topic

MoebiusFunction

Related topic

DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2

Related topic

DisplaystyleXlog2xOleftsum_nLeX2OmeganRight

Generated on Fri Feb 9 15:39:44 2018 by

LaTeXML
Dirichlet hyperbola method

Dirichlet hyperbola method

Let

f

,

g

, and

h

be

multiplicative functions

such that

f

=

g

*

h

, where

*

denotes the convolution (

http://planetmath.org/DirichletConvolution

) of

g

and

h

. The

Dirichlet hyperbola method

(typically shortened to

hyperbola method

) is a way to

∑

n

≤

x

f

⁢

(

n

)

by using the fact that

f

=

g

*

h

:

∑

n

≤

x

f

⁢

(

n

)

=

∑

n

≤

x

∑

a

⁢

b

=

n

g

⁢

(

a

)

⁢

h

⁢

(

b

)

=

∑

a

≤

x

∑

b

≤

x

a

g

⁢

(

a

)

⁢

h

⁢

(

b

)

+

∑

b

≤

x

∑

a

≤

x

b

g

⁢

(

a

)

⁢

h

⁢

(

b

)

-

∑

a

≤

x

∑

b

≤

x

g

⁢

(

a

)

⁢

h

⁢

(

b

)

Note that, since

a

⁢

b

=

n

≤

x

, not both of

a

and

b

can be larger than

x

. The Dirichlet hyperbola method follows from this fact as well as the

inclusion-exclusion principle

.

This method for calculating

∑

n

≤

x

f

⁢

(

n

)

is advantageous when the sums in of

g

and

h

are easier to handle and when

|

g

⁢

(

n

)

-

h

⁢

(

n

)

|

is relatively small for most

n

∈

ℕ

.

As an example, the sum

∑

n

≤

x

τ

⁢

(

n

)

will be calculated using the Dirichlet hyperbola method.

Note that

τ

=

1

*

1

. Thus:

∑

n

≤

x

τ

⁢

(

n

)

=

∑

a

≤

x

∑

b

≤

x

a

1

+

∑

b

≤

x

∑

a

≤

x

b

1

-

∑

a

≤

x

∑

b

≤

x

1

=

∑

a

≤

x

(

x

a

+

O

⁢

(

1

)

)

+

∑

b

≤

x

(

x

b

+

O

⁢

(

1

)

)

-

(

∑

a

≤

x

1

)

⁢

(

∑

b

≤

x

1

)

=

2

⁢

∑

c

≤

x

(

x

c

+

O

⁢

(

1

)

)

-

(

∑

c

≤

x

1

)

2

=

2

⁢

x

⁢

∑

c

≤

x

1

c

+

O

⁢

(

∑

c

≤

x

1

)

-

(

x

+

O

⁢

(

1

)

)

2

=

2

⁢

x

⁢

(

log

⁡

x

+

γ

+

O

⁢

(

1

x

)

)

+

O

⁢

(

x

)

-

(

x

+

O

⁢

(

x

)

+

O

⁢

(

1

)

)

=

2

⁢

x

⁢

(

1

2

⁢

log

⁡

x

+

γ

+

O

⁢

(

1

x

)

)

-

x

+

O

⁢

(

x

)

=

x

⁢

log

⁡

x

+

2

⁢

γ

⁢

x

+

O

⁢

(

x

x

)

-

x

+

O

⁢

(

x

)

=

x

⁢

log

⁡

x

+

(

2

⁢

γ

-

1

)

⁢

x

+

O

⁢

(

x

)

Title

Dirichlet hyperbola method

Canonical

name

DirichletHyperbolaMethod

Date of creation

2013-03-22 15:58:27

Last modified on

2013-03-22 15:58:27

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

14

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11N37

Synonym

hyperbola method

Related topic

ConvolutionMethod

Related topic

TauFunction

Related topic

EulersConstant

Generated on Fri Feb 9 15:39:48 2018 by

LaTeXML
Wirsing condition

Wirsing condition

Note that, within this entry,

p

always refers to a prime,

k

always refers to a

positive integer

, and

log

always refers to the

natural logarithm

.

Let

f

be a real-valued nonnegative

multiplicative function

. The

Wirsing condition

is that there exist

c

,

λ

∈

ℝ

with

c

≥

0

and

0

≤

λ

<

2

such that, for every prime

p

and every positive integer

k

,

f

⁢

(

p

k

)

≤

c

⁢

λ

k

.

The Wirsing condition is important because of the following lemma:

Lemma

.

If a real-valued nonnegative multiplicative function

f

the Wirsing condition, then it automatically the conditions in this theorem (

http://planetmath.org/AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions

). Those conditions are:

1.

There exists

A

≥

0

such that, for every

y

≥

0

,

∑

p

≤

y

f

⁢

(

p

)

⁢

log

⁡

p

≤

A

⁢

y

.

2.

There exists

B

≥

0

such that

∑

p

∑

k

≥

2

f

⁢

(

p

k

)

⁢

log

⁡

(

p

k

)

p

k

≤

B

.

Proof.

Let

f

the hypotheses of the lemma.

Let

y

≥

0

. Thus,

∑

p

≤

y

f

⁢

(

p

)

⁢

log

⁡

p

≤

c

⁢

λ

⁢

∑

p

≤

y

log

⁡

p

≤

c

⁢

λ

⁢

y

⁢

log

⁡

4

by this theorem (

http://planetmath.org/UpperBoundOnVarthetan

).

Also:

∑

p

∑

k

≥

2

f

⁢

(

p

k

)

⁢

log

⁡

(

p

k

)

p

k

≤

∑

p

∑

k

≥

2

c

⁢

λ

k

⋅

k

⁢

log

⁡

p

p

k

≤

c

⁢

∑

p

log

⁡

p

⁢

∑

k

≥

2

k

⁢

(

λ

p

)

k

≤

c

⁢

∑

p

log

⁡

p

⋅

2

⁢

(

λ

p

)

2

-

(

λ

p

)

3

(

1

-

λ

p

)

2

≤

2

⁢

c

(

1

-

λ

2

)

2

⁢

∑

p

log

⁡

p

⁢

(

λ

p

)

2

≤

2

⁢

c

⁢

λ

2

(

1

-

λ

2

)

2

⁢

∑

p

log

⁡

p

p

2

≤

2

⁢

c

⁢

λ

2

⁢

ζ

⁢

(

3

2

)

(

1

-

λ

2

)

2

, where

ζ

denotes the

Riemann zeta function

Hence,

A

=

c

⁢

λ

⁢

log

⁡

4

and

B

=

2

⁢

c

⁢

λ

2

⁢

ζ

⁢

(

3

2

)

(

1

-

λ

2

)

2

.
∎

Title

Wirsing condition

Canonical name

WirsingCondition

Date of creation

2013-03-22 16:08:45

Last modified on

2013-03-22 16:08:45

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11N37

Related topic

DisplaystyleSum_nLeXYomeganO_yxlogXy1ForYGe0

Generated on Fri Feb 9 15:39:51 2018 by

LaTeXML
∑@⁢ERROR(\slimits)⁢@⁢@⁢@_{n≤x}⁢y^{ω⁢(n)}=O_y⁢(x⁢(l⁢o⁢gx)^{y-1}) for y≥0

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

n

≤

x

⁢

y

ω

⁢

(

n

)

=

O

y

⁢

(

x

⁢

(

l

⁢

o

⁢

g

x

)

y

-

1

)

for

y

≥

0

Within this entry,

ω

refers to the

number of distinct prime factors function

,

⌊

⋅

⌋

refers to the

floor function

,

log

refers to the

natural logarithm

,

p

refers to a prime, and

k

and

n

refer to

positive integers

.

Theorem 1

.

For

y

≥

0

,

∑

n

≤

x

y

ω

⁢

(

n

)

=

O

y

⁢

(

x

⁢

(

log

⁡

x

)

y

-

1

)

.

Proof.

Since

y

ω

⁢

(

p

k

)

=

y

for all

p

and

k

, the real-valued nonnegative

multiplicative function

y

ω

⁢

(

n

)

the

Wirsing condition

with

c

=

y

and

λ

=

1

. Thus:

∑

n

≤

x

y

ω

⁢

(

n

)

=

O

y

⁢

(

x

log

⁡

x

⁢

∑

n

≤

x

y

ω

⁢

(

n

)

n

)

=

O

y

⁢

(

x

log

⁡

x

⁢

∏

p

≤

x

(

1

+

∑

k

=

1

⌊

log

⁡

x

log

⁡

p

⌋

y

ω

⁢

(

p

k

)

p

k

)

)

=

O

y

⁢

(

x

log

⁡

x

⁢

(

exp

⁡

(

∑

p

≤

x

∑

k

=

1

⌊

log

⁡

x

log

⁡

p

⌋

y

p

k

)

)

)

=

O

y

⁢

(

x

log

⁡

x

⁢

(

exp

⁡

(

y

⁢

∑

p

≤

x

∑

k

=

1

⌊

log

⁡

x

log

⁡

p

⌋

1

p

k

)

)

)

=

O

y

⁢

(

x

log

⁡

x

⁢

(

exp

⁡

(

y

⁢

(

log

⁡

(

log

⁡

x

)

+

O

⁢

(

1

)

)

)

)

)

=

O

y

(

x

log

⁡

x

(

exp

(

log

(

log

x

)

y

)

)

)

=

O

y

⁢

(

x

log

⁡

x

⁢

(

log

⁡

x

)

y

)

=

O

y

⁢

(

x

⁢

(

log

⁡

x

)

y

-

1

)

∎

Title

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

n

≤

x

⁢

y

ω

⁢

(

n

)

=

O

y

⁢

(

x

⁢

(

l

⁢

o

⁢

g

x

)

y

-

1

)

for

y

≥

0

Canonical name

displaystylesumnleXYomeganOyxlogXy1ForYge0

Date of creation

2013-03-22 16:11:22

Last modified on

2013-03-22 16:11:22

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11N37

Related topic

AsymptoticEstimate

Related topic

DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2

Related topic

WirsingCondition

Generated on Fri Feb 9 15:40:03 2018 by

LaTeXML
highly composite number

highly composite number

We call

n

a

highly composite number

if

d

⁢

(

n

)

>

d

⁢

(

m

)

for all

m

<

n

, where

d

⁢

(

n

)

is the number of divisors of

n

. The first several are 1, 2, 4, 6, 12, 24. The

sequence

is

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002182

A002182 in Sloane’s OEIS.

The

integer

n

is

superior

highly

composite

if there is an

ϵ

>

0

such that for all

m

≠

n

,

d

⁢

(

n

)

⁢

n

-

ϵ

>

d

⁢

(

m

)

⁢

m

-

ϵ

.

The first several superior highly composite numbers are 2, 6, 12, 60, 120, 360. The sequence is

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002201

A002201 in Sloane’s encyclopedia.

References

1

L. Alaoglu and P. Erdös,

On highly composite and similar numbers

. Trans. Amer. Math. Soc.

56

(1944), 448–469.

http://links.jstor.org/sici?sici=0002-9947%28194411%2956%3A3%3C448%3AOHCASN%3E2.0.CO%3B2-S

Available at www.jstor.org

Title

highly composite number

Canonical name

HighlyCompositeNumber

Date of creation

2013-03-22 13:40:44

Last modified on

2013-03-22 13:40:44

Owner

Kevin OBryant (1315)

Last modified by

Kevin OBryant (1315)

Numerical id

7

Author

Kevin OBryant (1315)

Entry type

Definition

Classification

msc 11N56

Defines

superior highly composite number

Generated on Fri Feb 9 15:40:10 2018 by

LaTeXML
list of divisors of some small highly composite numbers

list of divisors of some small highly composite numbers

The following table gives the

divisors

from 1 to

n

of

highly composite numbers

n

from 12 to 840. The number 1 is in the leftmost column, while the rightmost column (ignoring blank cells) gives

n

.

1

2

3

4

6

12

1

2

3

4

6

8

12

24

1

2

3

4

6

9

12

18

36

1

2

3

4

6

8

12

16

24

48

1

2

3

4

5

6

10

12

15

20

30

60

1

2

3

4

5

6

8

10

12

15

20

24

30

40

60

120

1

2

3

4

5

6

9

10

12

15

18

20

30

36

45

60

90

180

1

2

3

4

5

6

8

10

12

15

16

20

24

30

40

48

60

80

120

240

1

2

3

4

5

6

8

9

10

12

15

18

20

24

30

36

40

45

60

72

90

120

180

360

1

2

3

4

5

6

8

9

10

12

15

16

18

20

24

30

36

40

45

48

60

72

80

90

120

144

180

240

360

720

1

2

3

4

5

6

7

8

10

12

14

15

20

21

24

28

30

35

40

42

56

60

70

84

105

120

140

168

210

280

420

840

Title

list of divisors of some small highly composite numbers

Canonical name

ListOfDivisorsOfSomeSmallHighlyCompositeNumbers

Date of creation

2013-03-22 18:51:12

Last modified on

2013-03-22 18:51:12

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11N56

Generated on Fri Feb 9 15:40:12 2018 by

LaTeXML