aarjaneiro/planetmath huggingface dataset:
tetrahedral number

tetrahedral number

An

integer

of the form

(

n

2

+

n

)

⁢

(

n

+

2

)

6

,

where

n

is a

nonnegative integer

. Sometimes referred to as

T

n

,

tetrahedral numbers

are listed in A000292 of Sloane’s OEIS.

2

|

T

n

except when

n

≡

1

mod

4

.

With

t

n

the

n

th

triangular number

, the

n

th tetrahedral number can be calculated with this formula:

T

n

=

∑

i

=

1

n

t

i

.

Another way to calculate tetrahedral numbers is with the

binomial coefficient

T

n

=

(

n

+

2

3

)

.

This means that tetrahedral numbers can be looked up in Pascal’s triangle.

Tetrahedral numbers have practical applications in sphere packing.

Title

tetrahedral number

Canonical name

TetrahedralNumber

Date of creation

2013-03-22 15:56:34

Last modified on

2013-03-22 15:56:34

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A99

Synonym

triangular pyramidal number

Generated on Fri Feb 9 15:25:23 2018 by

LaTeXML
thirteen

thirteen

Thirteen

is in many ways a rather ordinary number, yet some people are afraid of it (see

triskaidekaphobia

). For a mathematician there is nothing particularly unlucky about 13; in fact, 13 is a

lucky prime

(since it is not crossed off in the

well-defined

sieving process for

lucky numbers

).

13 is a

Fibonacci number

(being the sum of 5 and 8), and like all other odd-indexed Fibonacci numbers, it is also a

Markov number

. It appears in the following solutions to

13

2

+

y

2

+

z

2

=

39

⁢

y

⁢

z

: (1, 5, 13), (1, 13, 34), (5, 13, 194), (13, 34, 1325), (13, 194, 7561), (13, 1325, 51641), (13, 7561, 294685), (13, 51641, 2012674), (13, 294685, 11485154), (13, 2012674, 78442645), (13, 11485154, 447626321), etc. Starting with 1, the number 13 represents a new low for the

Mertens function

, but this is neither unlucky nor special (A051401 in Sloane’s OEIS lists other lows of the Mertens function).

This is not to say that there isn’t anything special about 13: for example, it is the only

integer

solution to

x

4

=

n

2

+

(

n

+

1

)

2

(besides the

obvious

x

=

1

and

n

=

0

).

Title

thirteen

Canonical

name

Thirteen

Date of creation

2013-03-22 17:24:58

Last modified on

2013-03-22 17:24:58

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Feature

Classification

msc 11A99

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

LaTeXML
triangular numbers

triangular numbers

The

triangular numbers

are defined by the series

t

n

=

∑

i

=

1

n

i

That is, the

n

th triangular number is simply the sum of the first

n

natural numbers

. The first few triangular numbers are

1

,

3

,

6

,

10

,

15

,

21

,

28

,

…

The name triangular number comes from the fact that the

summation

defining

t

n

can be visualized as the number of dots in

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

⋮

⋮

⋱

where the number of rows is equal to

n

.

The

closed-form

for the triangular numbers is

t

⁢

(

n

)

=

n

⁢

(

n

+

1

)

2

Legend has it that a grammar-school-aged Gauss was told by his teacher to sum up all the numbers from 1 to 100. He reasoned that each number

i

could be paired up with

101

-

i

, to form a sum of

101

, and if this was done

100

times, it would result in twice the actual sum (since each number would get used twice due to the

pairing

). Hence, the sum would be

1

+

2

+

3

+

⋯

+

100

=

100

⁢

(

101

)

2

The same line of reasoning works to give us the closed form for any

n

.

Another way to derive the closed form is to assume that the

n

th triangular number is less than or equal to the

n

th square (that is, each row is less than or equal to

n

, so the sum of all rows must be less than or equal to

n

⋅

n

or

n

2

), and then use the first few triangular numbers to solve the general 2nd degree

polynomial

A

⁢

n

2

+

B

⁢

n

+

C

for

A

,

B

, and

C

. This leads to

A

=

1

/

2

,

B

=

1

/

2

, and

C

=

0

, which is the same as the above

formula

for

t

⁢

(

n

)

.

Title

triangular numbers

Canonical

name

TriangularNumbers

Date of creation

2013-03-22 12:16:08

Last modified on

2013-03-22 12:16:08

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

6

Author

akrowne (2)

Entry type

Definition

Classification

msc 11A99

Classification

msc 40-00

Generated on Fri Feb 9 15:25:27 2018 by

LaTeXML
two hundred thirty-nine

two hundred thirty-nine

The MIT

Artificial Intelligence

Laboratory Memo 239 (entitled

HAKMEM

) of February 1972 listed various properties of the

integer

239, some more interesting than others, such as: that 239 needs the

maximum

number of powers in Waring’s problem for squares, cubes and

fourth powers

; that it appears in Machin’s

formula

for

π

:

1

4

⁢

π

=

4

⁢

cot

-

1

⁡

5

-

cot

-

1

⁡

239

;

that since

239

2

=

2

×

13

4

-

1

, the

fraction

239

13

2

is a

convergent

for the

continued fraction

of

2

; that since

π

⁢

(

1500

)

=

239

(with

π

⁢

(

x

)

being the

prime counting function

; etc. At the time, the memo couldn’t have mentioned that 239 pounds is Homer Simpson’s weight (as established by at least two episodes of

The Simpsons

).

The prime 239, like many in its vicinity, is a

Chen prime

. As a

Sophie Germain prime

, it begins a

Cunningham chain

of just length 2 (which ends with 479). On the plane of

Eisenstein integers

, 239 is an

Eisenstein prime

(its

real part

being of the form

3

⁢

n

-

1

and it not having an

imaginary part

), and it is also a

Gaussian prime

(its real part being of the form

4

⁢

n

-

1

); these two properties it has in common with all

real primes

p

≡

11

mod

12

. Much more rare is that it is the third Newman-Shanks-Williams prime (the ninth is more than

10

98

).

Title

two hundred thirty-nine

Canonical name

TwoHundredThirtynine

Date of creation

2013-03-22 16:57:56

Last modified on

2013-03-22 16:57:56

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Feature

Classification

msc 11A99

Synonym

two hundred and thirty-nine

Generated on Fri Feb 9 15:25:29 2018 by

LaTeXML
ultimate generalisation of Euler-Fermat theorem

ultimate generalisation of Euler-Fermat theorem

Let

a

b

+

u

=

m

where

a

,

b

,

u

,

m

are

positive integers

. Then

a

b

+

k

⁢

φ

⁢

(

m

)

+

u

≡

0

(

mod

m

)

,

by the result in “Euler’s generalisation of Fermat’s

theorem

– a further generalisation”.

Proceedings of Hawaii Intl. conference on maths & statistics

2004

(ISSN 1550–3747). Here,

k

is a positive integer. Next,

a

b

1

+

k

⁢

φ

⁢

(

φ

⁢

(

m

)

)

+

u

≡

0

(

mod

m

)

.

(This is a corollary of “Euler’s generalisation of Fermat’s theorem – a further generalisation”.)
We can proceed in a like manner till we reach

a

b

c

⋮

t

1

+

k

⁢

φ

⁢

(

φ

⁢

(

φ

⁢

(

…

⁢

φ

⁢

(

2

)

⁢

…

)

)

)

.

At this stage onwards the

function

generates only

multiples

of

m

and no

prime number

is generated. This is the ultimate generalisation of Fermat’s theorem. Please note that each step of multiple

exponentiation

in the above is a corollary of the theorem referred to.

Title

ultimate generalisation of Euler-Fermat theorem

Canonical name

UltimateGeneralisationOfEulerFermatTheorem

Date of creation

2013-03-22 19:35:04

Last modified on

2013-03-22 19:35:04

Owner

akdevaraj (13230)

Last modified by

akdevaraj (13230)

Numerical id

5

Author

akdevaraj (13230)

Classification

msc 11A99

Generated on Fri Feb 9 15:25:31 2018 by

LaTeXML
asymptotic density

asymptotic density

Let

A

be a subset of

ℤ

+

. For any

n

∈

ℤ

+

put

A

⁢

(

n

)

=

{

1

,

2

,

…

,

n

}

∩

A

.

Define the

upper asymptotic density

d

¯

⁢

(

A

)

of

A

by

d

¯

⁢

(

A

)

=

lim sup

n

→

∞

⁡

|

A

⁢

(

n

)

|

n

d

¯

⁢

(

A

)

is also known simply as the

upper density

of

A

.

Similarly, we define

d

¯

⁢

(

A

)

, the

lower asymptotic density

of

A

, by

d

¯

⁢

(

A

)

=

lim inf

n

→

∞

⁡

|

A

⁢

(

n

)

|

n

We say

A

has

asymptotic density

d

⁢

(

A

)

if

d

¯

⁢

(

A

)

=

d

¯

⁢

(

A

)

, in which case we put

d

⁢

(

A

)

=

d

¯

⁢

(

A

)

.

Title

asymptotic density

Canonical

name

AsymptoticDensity

Date of creation

2013-03-22 12:36:20

Last modified on

2013-03-22 12:36:20

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11B05

Synonym

upper density

Synonym

lower density

Synonym

natural density

Synonym

arithmetic density

Related topic

InequalityOfLogarithmicAndAsymptoticDensity

Defines

upper asymptotic density

Defines

lower asymptotic density

Generated on Fri Feb 9 15:25:32 2018 by

LaTeXML
Cauchy-Davenport theorem

Cauchy-Davenport theorem

If

A

and

B

are non-empty subsets of

ℤ

p

, then

|

A

+

B

|

≥

min

⁡

(

|

A

|

+

|

B

|

-

1

,

p

)

,

where

A

+

B

denotes the sumset of

A

and

B

.

References

1

Melvyn B. Nathanson.

Additive Number Theory

:

Inverse Problems

and Geometry of
Sumsets

, volume 165 of

GTM

.

Springer, 1996.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003

Zbl 0859.11003.

Title

Cauchy-Davenport theorem

Canonical name

CauchyDavenportTheorem

Date of creation

2013-03-22 13:32:13

Last modified on

2013-03-22 13:32:13

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11B05

Related topic

Sumset

Generated on Fri Feb 9 15:25:34 2018 by

LaTeXML
essential component

essential component

If

A

is a set of nonnegative integers such that

σ

⁢

(

A

+

B

)

>

σ

⁢

B

(1)

for every set

B

with

Schnirelmann density

0

<

σ

⁢

B

<

1

, then

A

is an

essential component

.

Erdős proved that every

http://planetmath.org/node/3831

basis is an essential component. In fact he proved that

σ

⁢

(

A

+

B

)

≥

σ

⁢

B

+

1

2

⁢

h

⁢

(

1

-

σ

⁢

B

)

⁢

σ

⁢

B

,

where

h

denotes the

http://planetmath.org/node/3831

order of

A

.

Plünnecke improved that to

σ

⁢

(

A

+

B

)

≥

σ

⁢

B

1

-

1

/

h

.

There are non-basic essential components. Linnik constructed non-basic essential component for which

A

⁢

(

n

)

=

O

⁢

(

n

ϵ

)

for every

ϵ

>

0

.

References

1

Heini Halberstam and Klaus Friedrich Roth.

Sequences

.

Springer-Verlag, second edition, 1983.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0498.10001

Zbl 0498.10001.

Title

essential component

Canonical name

EssentialComponent

Date of creation

2013-03-22 13:19:42

Last modified on

2013-03-22 13:19:42

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Definition

Classification

msc 11B05

Classification

msc 11B13

Related topic

SchnirlemannDensity

Related topic

Basis2

Generated on Fri Feb 9 15:25:36 2018 by

LaTeXML
logarithmic density

logarithmic density

For any

A

⊆

ℕ

we denote

S

⁢

(

n

)

:=

∑

k

=

1

n

1

k

. The values

δ

¯

⁢

(

A

)

=

lim inf

n

→

∞

⁡

∑

k

∈

A

;

k

≤

n

1

k

S

⁢

(

n

)

δ

¯

⁢

(

A

)

=

lim sup

n

→

∞

⁡

∑

k

∈

A

;

k

≤

n

1

k

S

⁢

(

n

)

are called

lower

and

upper logarithmic density

of

A

. If

δ

¯

⁢

(

A

)

=

δ

¯

⁢

(

A

)

we denote this value by

δ

⁢

(

A

)

and call it the

logarithmic density

of

A

.

Logarithmic density can be equivalently defined as follows: If the limit

δ

⁢

(

A

)

=

lim

n

→

∞

⁡

∑

k

∈

A

;

k

≤

n

1

k

S

⁢

(

n

)

,

exists, then it is called logarithmic density of

A

.

By the well-known

γ

=

lim

n

→

∞

⁡

S

⁢

(

n

)

-

ln

⁡

n

defining Euler’s

constant

, we can see that the

denominator

in the
above definitions can be replaced by

ln

⁡

n

.

References

1

M. Kolibiar, A. Legéň, T. Šalát, and Š. Znám.

Algebra a príbuzné disciplíny

.

Alfa, Bratislava, 1992. (in Slovak)

2

H. H. Ostmann.

Additive

Zahlentheorie I

.

Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956.

3

J. Steuding.

http://www.math.uni-frankfurt.de/ steuding/steuding/prob.pdf

Probabilistic number theory.

4

G. Tenenbaum.

Introduction

to

analytic

and probabilistic number theory

.

Cambridge Univ. Press, Cambridge, 1995.

Title

logarithmic density

Canonical name

LogarithmicDensity

Date of creation

2013-03-22 15:31:54

Last modified on

2013-03-22 15:31:54

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

6

Author

kompik (10588)

Entry type

Definition

Classification

msc 11B05

Related topic

InequalityOfLogarithmicAndAsymptoticDensity

Defines

upper logarithmic density

Defines

lower logarithmic density

Generated on Fri Feb 9 15:25:42 2018 by

LaTeXML
Mann’s theorem

Mann’s theorem

Let

A

and

B

be subsets of

ℤ

. If

0

∈

A

∩

B

,

σ

⁢

(

A

+

B

)

≥

min

⁡

(

1

,

σ

⁢

A

+

σ

⁢

B

)

,

where

σ

denotes

Schnirelmann density

.

This statement was known also as

(

α

+

β

)

-

conjecture

until H. B. Mann proved it in 1942.

Title

Mann’s

theorem

Canonical

name

MannsTheorem

Date of creation

2013-03-22 13:20:32

Last modified on

2013-03-22 13:20:32

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

5

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11B05

Classification

msc 11B13

Synonym

(

α

+

β

)

-conjecture

Related topic

SchnirlemannDensity

Generated on Fri Feb 9 15:25:44 2018 by

LaTeXML
normal order

normal order

Let

f

⁢

(

n

)

and

F

⁢

(

n

)

be functions from

ℤ

+

→

ℝ

. We say that

f

⁢

(

n

)

has

normal order

F

⁢

(

n

)

if for each

ϵ

>

0

the
set

A

⁢

(

ϵ

)

=

{

n

∈

ℤ

+

:

(

1

-

ϵ

)

⁢

F

⁢

(

n

)

<

f

⁢

(

n

)

<

(

1

+

ϵ

)

⁢

F

⁢

(

n

)

}

has the

property

that

d

¯

⁢

(

A

⁢

(

ϵ

)

)

=

1

.
Equivalently, if

B

⁢

(

ϵ

)

=

ℤ

+

\

A

⁢

(

ϵ

)

, then

d

¯

⁢

(

B

⁢

(

ϵ

)

)

=

0

. (Note that

d

¯

⁢

(

X

)

denotes the

lower asymptotic density

of

X

).

We say that

f

has

average order

F

if

∑

j

=

1

n

f

⁢

(

j

)

∼

∑

j

=

1

n

F

⁢

(

j

)

Title

normal order

Canonical

name

NormalOrder

Date of creation

2013-03-22 12:36:23

Last modified on

2013-03-22 12:36:23

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11B05

Defines

average order

Generated on Fri Feb 9 15:25:46 2018 by

LaTeXML
proof of Cauchy-Davenport theorem

proof of Cauchy-Davenport theorem

There is a proof, essentially from here

http://www.math.tau.ac.il/ nogaa/PDFS/annr3.pdf

(Imre Ruzsa et al.):

Let

|

A

|

=

k

and

|

B

|

=

l

and

|

A

+

B

|

=

n

.
If

n

≥

k

+

l

-

1

, the assertion is true, now assume

n

<

k

+

l

-

1

.
Form the

polynomial

f

⁢

(

x

,

y

)

=

∏

c

∈

A

+

B

(

x

+

y

-

c

)

=

∑

i

+

j

≤

n

f

i

⁢

j

*

x

i

*

y

j

.
The sum is over

i

,

j

with

i

+

j

≤

n

, because there are

n

factors in the

product

.

Since

Z

p

is a field, there are polynomials

g

i

of degree

<

k

and

h

j

of degree

<

l

such that

g

i

⁢

(

x

)

=

x

i

for all

x

∈

A

and

h

j

⁢

(

y

)

=

y

j

for all

y

∈

B

.
Define a polynomial

p

by

p

⁢

(

x

,

y

)

=

∑

i

<

k

,

j

<

l

f

i

⁢

j

*

x

i

*

y

j

+

∑

i

≥

k

,

j

≤

n

-

i

f

i

⁢

j

*

g

i

⁢

(

x

)

*

y

j

+

∑

j

≥

l

,

i

≤

n

-

j

f

i

⁢

j

*

x

i

*

h

j

⁢

(

y

)

.

This polynomial coincides with

f

⁢

(

x

,

y

)

for all

x

in

A

and

y

in

B

, for these

x

,

y

we have, however,

f

⁢

(

x

,

y

)

=

0

.
The polynomial

p

⁢

(

x

,

y

)

is of degree

<

k

in

x

and of degree

<

l

in

y

.
Let

x

∈

A

, then

p

⁢

(

x

,

y

)

=

∑

p

j

⁢

(

x

)

*

y

j

is zero for all

y

∈

B

, and all coefficients must be zero. Finally,

p

j

⁢

(

x

)

is zero for all

x

∈

A

, and all coefficients

p

i

⁢

j

of

p

⁢

(

x

,

y

)

=

∑

p

i

⁢

j

*

x

i

*

y

j

must be zero as

elements of

Z

p

.

Should the assertion of the

theorem

be false, then there are numbers

u

,

v

with

u

+

v

=

n

<

p

and

u

<

k

and

v

<

l

.

But the monomial

x

u

*

y

v

does not appear in the second and third sum, because for

j

=

v

we have

i

≤

n

-

j

=

u

<

k

, and for

i

=

u

we have

j

≤

n

-

i

=

v

<

l

.
Then

p

u

⁢

v

=

0

is

m

⁢

o

⁢

d

⁢

u

⁢

l

⁢

o

⁢

p

equal to

f

u

⁢

v

, this is equal to the

binomial coefficient

(

n

v

)

, which is not

divisible

by

p

for

n

<

p

, a

contradiction

.
The

Cauchy-Davenport theorem

is proved.

Title

proof of Cauchy-Davenport theorem

Canonical name

ProofOfCauchyDavenportTheorem

Date of creation

2013-03-22 14:34:49

Last modified on

2013-03-22 14:34:49

Owner

Wolfgang (5320)

Last modified by

Wolfgang (5320)

Numerical id

26

Author

Wolfgang (5320)

Entry type

Proof

Classification

msc 11B05

Generated on Fri Feb 9 15:25:49 2018 by

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addition chain

addition chain

An

addition chain

a

is a

sequence

of

integers

of length

k

such that each term

a

i

for

0

<

i

≤

k

(with

a

0

=

1

) is the sum of two previous terms in at least one way. In the sum

a

m

+

a

n

it is not required that

m

≠

n

. For example, 1, 2, 3, 5, 10, 20, 40, 80, is an addition chain of length 7: 3 is 1 + 2, 5 = 2 + 3, 10 = 5 + 5, and the rest have

m

=

n

.

There are various subclassifications of addition chains, such as the

Lucas chains

. A

Mian-Chowla sequence

is an addition chain with the

restriction

that each term is the sum of two previous terms in only one way. The length may be

infinite

, and thus the

Fibonacci sequence

is an addition chain.

Title

addition chain

Canonical name

AdditionChain

Date of creation

2013-03-22 18:27:27

Last modified on

2013-03-22 18:27:27

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B13

Generated on Fri Feb 9 15:25:53 2018 by

LaTeXML
additive basis

additive basis

A subset

A

of

ℤ

is an

() basis

of

n

if

n

⁢

A

=

ℕ

∪

{

0

}

,

where

n

⁢

A

is

n

-fold sumset of

A

. Usually it is assumed that

0

belongs to

A

when saying that

A

is an

additive basis

.

Title

additive basis

Canonical name

AdditiveBasis

Date of creation

2013-03-22 13:19:17

Last modified on

2013-03-22 13:19:17

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Definition

Classification

msc 11B13

Synonym

basis

Related topic

AsymptoticBasis

Related topic

SchnirlemannDensity

Related topic

EssentialComponent

Defines

order of additive basis

Generated on Fri Feb 9 15:25:55 2018 by

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asymptotic basis

asymptotic basis

A subset

A

of

ℤ

is an

asymptotic basis

of

n

if the

n

-fold sumset

n

⁢

A

contains all sufficiently large

integers

.

Title

asymptotic basis

Canonical name

AsymptoticBasis

Date of creation

2013-03-22 13:19:19

Last modified on

2013-03-22 13:19:19

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

4

Author

bbukh (348)

Entry type

Definition

Classification

msc 11B13

Related topic

Basis2

Defines

order of asymptotic basis

Generated on Fri Feb 9 15:25:56 2018 by

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Erdős-Turan conjecture

Erdős-Turan conjecture

Erdős-Turan conjecture asserts there exist no

asymptotic basis

(

http://planetmath.org/AsymptoticBasis

)

A

⊂

ℕ

0

of order

2

such that its representation function

r

A

,

2

′

⁢

(

n

)

=

∑

a

1

+

a

2

=

n

a

1

≤

a

2

1

is

bounded

.

Alternatively, the question can be phrased as whether there exists a

power series

F

with coefficients

0

and

1

such that all coefficients of

F

2

are greater than

0

, but are bounded.

If we replace set of nonnegative integers by the

set of all integers

, then the question was settled by Nathanson

[

2

]

in

negative

, that is, there exists a set

A

⊂

ℤ

such that

r

A

,

2

′

⁢

(

n

)

=

1

.

References

1

Heini Halberstam and Klaus Friedrich Roth.

Sequences

.

Springer-Verlag, second edition, 1983.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0498.10001

Zbl 0498.10001.

2

Melvyn B. Nathanson.

Every

function

is the representation function of an

additive basis

for the

integers

.

http://front.math.ucdavis.edu/math.NT/0302091

arXiv:math.NT/0302091.

Title

Erdős-Turan conjecture

Canonical name

ErdHosTuranConjecture

Date of creation

2013-03-22 13:27:11

Last modified on

2013-03-22 13:27:11

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

8

Author

bbukh (348)

Entry type

Conjecture

Classification

msc 11B13

Classification

msc 11B34

Classification

msc 11B05

Related topic

SidonSet

Generated on Fri Feb 9 15:26:03 2018 by

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Euler’s conjecture

Euler’s conjecture

In 1769, Leonhard Euler conjectured that for

1

<

k

<

n

there is no set of positive integers

a

1

,

…

,

a

k

such that

∑

i

=

1

k

a

i

n

=

m

n

,

where

m

is an

integer

. Lander and Parkin in 1966 disproved the

conjecture

with this

k

=

4

,

n

=

5

counterexample

:

27

5

+

84

5

+

110

5

+

133

5

=

144

5

. More counterexamples have been discovered since then.

Title

Euler’s conjecture

Canonical

name

EulersConjecture

Date of creation

2013-03-22 16:25:43

Last modified on

2013-03-22 16:25:43

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11B13

Classification

msc 11D41

Synonym

Euler conjecture

Generated on Fri Feb 9 15:26:05 2018 by

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eventually coincide

eventually coincide

Let

A

and

B

be two

nonempty sets

of

integers

. We say that

A

and

B

eventually coincide

if there is an integer

C

such that

n

∈

A

if and only if

n

∈

B

for all

n

≥

C

. In this case, we write

A

∼

B

, noting that the

relation

of eventually coinciding is clearly an

equivalence relation

. While a seemingly

trivial

notation, this turns out to be the “right” notion of of sets when dealing with asymptotic

properties

such as .

References

1

Nathanson, Melvyn B.,

Elementary Methods

in

Number Theory

, Graduate Texts in Mathematics, Volume 195. Springer-Verlag, 2000.

Title

eventually coincide

Canonical

name

EventuallyCoincide

Date of creation

2013-03-22 15:09:30

Last modified on

2013-03-22 15:09:30

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11B13

Synonym

eventually coinciding

Generated on Fri Feb 9 15:26:07 2018 by

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Lucas chain

Lucas chain

A

Lucas chain

a

is an

addition chain

with the additional requirement that not only each term be the sum of two previous (not necessarily distinct) terms, but also that the

difference

of those two terms also be a term in the

sequence

. That is, each

a

i

=

a

m

+

a

n

and also

|

a

n

-

a

m

|

=

a

j

, with

j

being some

nonnegative integer

.

For example, the

Fibonacci sequence

(1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.) is a Lucas chain because not only is each term the sum of the previous two terms, each term is the difference of the next two terms. On the other hand, 1, 2, 4, 8, 9, 13, 21, 30, etc., is an addition chain but not a Lucas chain, since 8 + 13 = 21, but

13

-

8

=

5

, which is not a member of the chain.

Title

Lucas chain

Canonical

name

LucasChain

Date of creation

2013-03-22 18:29:22

Last modified on

2013-03-22 18:29:22

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B13

Generated on Fri Feb 9 15:26:09 2018 by

LaTeXML