aarjaneiro/planetmath huggingface dataset:
examples supporting the Erdős-Straus conjecture

examples supporting the Erdős-Straus conjecture

As with any

conjecture

, a million examples are not enough to prove the Erdős-Straus conjecture, but a single

counterexample

is enough to disprove. But if these examples at least provide a slightly better understanding of the problem at hand, the effort is not entirely wasted.

Most users are well aware that a

computer algebra system

(and even fraction-capable

scientific calculators

) will automatically express a

fraction

by the lowest common denominator. This presents no problem for smaller instances, such as

4

8

, but for larger denominators it might not always be

obvious

that, for example,

2

1729

=

4

3458

. If one is unsure, one can always enter the fraction

4

n

by itself and the CAS will dutifully respond with the LCD

expression

, which will hopefully match the sum of three

unit fractions

entered earlier.

I say we start with

n

=

2

if for no other reason than to start at the beginning. The only possible

solution

is

4

2

=

1

+

1

2

+

1

2

.

Similarly, for

n

=

3

we have

4

3

=

1

2

+

1

2

+

1

3

.

It is with

n

=

4

that solutions with distinct denominators become available:

4

4

=

1

2

+

1

3

+

1

6

,

easily suggested by the study of

perfect numbers

. Some may consider solutions with distinct denominators more elegant, some are just happy to find any solution for a given

n

.

Since

addition

is

commutative

, it does not matter in what order we list the unit fractions that add up to our desired

4

n

. However, by tradition, they are listed in

descending order

: the biggest fraction (the one with the smallest denominator) is listed first, the smallest last.

The following table lists some distinct denominator solutions for

4

<

n

<

21

, with the numerators omitted for compactness:

5

2, 5, 10

6

2, 8, 24; 3, 4, 12

7

2, 21, 42; 3, 6, 14

8

3, 6, 42; 3, 8, 24; 4, 6, 12

9

3, 12, 36; 4, 6, 36

10

3, 20, 60; 5, 6, 30; 4, 10, 20

11

3, 44, 132; 4, 11, 44; 4, 12, 33

12

4, 18, 36; 4, 16, 48; 6, 8, 24

13

4, 26, 52

14

4, 42, 84; 5, 70, 140; 6, 14, 21; 7, 8, 56; 6, 12, 28

15

4, 90, 180; 5, 18, 90; 6, 15, 30; 7, 10, 42

16

5, 30, 60; 6, 12, 84; 8, 12, 24; 6, 20, 30

17

5, 30, 510; 6, 17, 102

18

6, 24, 72; 8, 12, 72

19

6, 38, 57

20

7, 20, 140

As the table shows, solutions for prime

n

are harder to come by than for

composite

n

. When

n

=

p

⁢

q

, with

p

prime and

q

any other

integer

, solutions for

n

can be simply derived from those for

p

by multiplying those denominators by

q

. For example, for

n

=

42

, we can take the solutions for

n

=

6

, multiply those denominators by 7 and voilà:

4

42

=

1

12

+

1

126

+

1

252

.

Title

examples supporting the Erdős-Straus conjecture

Canonical

name

ExamplesSupportingTheErdHosStrausConjecture

Date of creation

2013-03-22 16:54:10

Last modified on

2013-03-22 16:54:10

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

9

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A67

Generated on Fri Feb 9 15:24:37 2018 by

LaTeXML
sums of two squares

sums of two squares

Theorem

.

The set of the

sums of two squares

of

integers

is

closed under

multiplication

; in fact we have the identical equation

(

a

2

+

b

2

)

⁢

(

c

2

+

d

2

)

=

(

a

⁢

c

-

b

⁢

d

)

2

+

(

a

⁢

d

+

b

⁢

c

)

2

.

(1)

This was presented by Leonardo Fibonacci in 1225 (in

Liber quadratorum

), but was known also by Brahmagupta
and already by Diophantus of Alexandria (III book of his

Arithmetica

).

The proof of the equation may utilize

Gaussian integers

as follows:

(

a

2

+

b

2

)

⁢

(

c

2

+

d

2

)

=

(

a

+

i

⁢

b

)

⁢

(

a

-

i

⁢

b

)

⁢

(

c

+

i

⁢

d

)

⁢

(

c

-

i

⁢

d

)

=

(

a

+

i

⁢

b

)

⁢

(

c

+

i

⁢

d

)

⁢

(

a

-

i

⁢

b

)

⁢

(

c

-

i

⁢

d

)

=

[

(

a

⁢

c

-

b

⁢

d

)

+

i

⁢

(

a

⁢

d

+

b

⁢

c

)

]

⁢

[

(

a

⁢

c

-

b

⁢

d

)

-

i

⁢

(

a

⁢

d

+

b

⁢

c

)

]

=

(

a

⁢

c

-

b

⁢

d

)

2

+

(

a

⁢

d

+

b

⁢

c

)

2

Note 1.

The equation (1) is the special case

n

=

2

of Lagrange’s

identity

.

Note 2.

Similarly as (1), one can derive the identity

(

a

2

+

b

2

)

⁢

(

c

2

+

d

2

)

=

(

a

⁢

c

+

b

⁢

d

)

2

+

(

a

⁢

d

-

b

⁢

c

)

2

.

(2)

Thus in most cases, we can get two different nontrivial sum forms
(i.e. without a zero addend) for a given

product

of two sums of
squares.  For example, the product

65

=

5

⋅

13

=

(

2

2

+

1

2

)

⁢

(

3

2

+

2

2

)

attains the two forms

4

2

+

7

2

and

8

2

+

1

2

.

Title

sums of two squares

Canonical

name

SumsOfTwoSquares

Date of creation

2013-11-19 16:28:21

Last modified on

2013-11-19 16:28:21

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

33

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A67

Classification

msc 11E25

Synonym

Diophantus’ identity

Synonym

Brahmagupta’s identity

Synonym

Fibonacci’s identity

Related topic

EulerFourSquareIdentity

Related topic

TheoremsOnSumsOfSquares

Related topic

DifferenceOfSquares

Generated on Fri Feb 9 15:24:52 2018 by

LaTeXML
unit fraction

unit fraction

An

unit fraction

n

d

is a

fraction

whose numerator

n

=

1

.
If its

integer

denominator

d

>

1

, then a fraction is also a proper fraction. So there is only one unit fraction which is improper, namely 1.

Such fractions are known from

Egyptian mathematics

where we can find a lot of special representations of the numbers as a sum of an unit fractions, which are now called

Egyptian fractions

. From the

Rhind papyrus

as an example:

2

71

=

1

40

+

1

568

+

1

710

.

Many unit fractions are in the pairs of the

adjacent fractions

. An unit fractions are some successive or non-successive terms of any

Farey sequence

F

n

of a degree

n

. For example the fractions

1

2

and

1

4

are

adjacent

, but they are not the successive terms in the Farey sequence

F

5

. The fractions

1

3

and

1

4

are also adjacent and they are successive terms in the

F

5

.

Title

unit fraction

Canonical name

UnitFraction

Date of creation

2013-03-22 12:48:25

Last modified on

2013-03-22 12:48:25

Owner

XJamRastafire (349)

Last modified by

XJamRastafire (349)

Numerical id

10

Author

XJamRastafire (349)

Entry type

Definition

Classification

msc 11A67

Related topic

AdjacentFraction

Related topic

AnyRationalNumberIsASumOfUnitFractions

Related topic

AnyRationalNumberWithOddDenominatorIsASumOfUnitFractionsWithOddDenominators

Related topic

UnitFraction

Related topic

SierpinskiErdosEgyptianFractionConjecture

Defines

Egyptian fraction

Generated on Fri Feb 9 15:24:54 2018 by

LaTeXML
ABC conjecture

ABC conjecture

The

ABC conjecture

states that given any

ϵ

>

0

,
there is a constant

κ

⁢

(

ϵ

)

such that

max

⁡

(

|

A

|

,

|

B

|

,

|

C

|

)

≤

κ

⁢

(

ϵ

)

⁢

(

rad

⁡

(

A

⁢

B

⁢

C

)

)

1

+

ϵ

for all

mutually coprime

integers

A

,

B

,

C

with

A

+

B

=

C

,
where

rad

is the

radical

of an integer.
This

conjecture

was formulated by Masser and Oesterlé in 1980.

The ABC conjecture is considered
one of the most important unsolved problems in number ,
as many results would follow directly from this conjecture.
For example, Fermat’s Last

Theorem

could be proved (for sufficiently large exponents)
with about one page worth of proof.

Further Reading

http://www.maa.org/mathland/mathtrek_12_8.html

The Amazing ABC Conjecture
— an article on the ABC conjecture by Ivars Peterson.

http://www.hcs.harvard.edu/hcmr/issue1/elkies.pdf

The ABC’s of

Number Theory

— an article on the ABC conjecture by Noam Elkies. (PDF file)

Title

ABC conjecture

Canonical

name

ABCConjecture

Date of creation

2013-03-22 11:45:23

Last modified on

2013-03-22 11:45:23

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

21

Author

yark (2760)

Entry type

Conjecture

Classification

msc 11A99

Classification

msc 55-00

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Classification

msc 18-00

Classification

msc 18C10

Generated on Fri Feb 9 15:24:55 2018 by

LaTeXML
converting a repeating decimal to a fraction

converting a repeating decimal to a fraction

The following

algorithm

can be used to convert a repeating decimal to a fraction:

1.

Set the repeating decimal equal to

x

.

2.

Multiply both sides of the equation by

10

n

, where

n

is the number of digits that appear under the bar.

3.

If applicable, rewrite the second equation so that its repeating part up with the repeating part in the original equation.

4.

Subtract the original equation from the most recently obtained equation. (The repeating part should cancel at this step.)

5.

If applicable, multiply both sides by a large enough power of

10

so that the equation is of the form

a

⁢

x

=

b

, where

a

and

b

are integers.

6.

Divide both sides of the equation by the coefficient of

x

.

7.

Reduce the fraction to lowest terms.

Below, this algorithm is demonstrated for

0.58

⁢

3

¯

with the steps indicated on the far .

x

=

0.58

⁢

3

¯

(1)

10

⁢

x

=

5.8

⁢

3

¯

(2)

10

⁢

x

=

5.83

⁢

3

¯

(3)

9

⁢

x

=

5.25

(4)

900

⁢

x

=

525

(5)

x

=

525

900

(6)

x

=

7

12

(7)

An important application of this algorithm is that it supplies a proof for the fact that

0

.

9

¯

=

1

:

x

=

0

.

9

¯

10

⁢

x

=

9

.

9

¯

9

⁢

x

=

9

x

=

1

Title

converting a repeating decimal to a fraction

Canonical name

ConvertingARepeatingDecimalToAFraction

Date of creation

2013-03-22 16:55:22

Last modified on

2013-03-22 16:55:22

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Algorithm

Classification

msc 11A99

Classification

msc 11-00

Generated on Fri Feb 9 15:24:58 2018 by

LaTeXML
forty-two

forty-two

Ever since Douglas Adams’s

Hitchhiker’s Guide to the Galaxy

was first published, amateur mathematicians have sought to give explanations, with varying degrees of seriousness, as to why

forty-two

is

The Answer to Life, the

Universe

, and Everything

.

In the Adams book, 42 is the answer given by the computer Deep Thought, which then tells the people waiting to hear the answer that they don’t know what the question is. Arthur Dent tries to assemble the question from Scrabble tiles, coming up with the question ”What do you get if you multiply six by nine?” The answer is

54

=

42

13

. Adams says he did not intend to reference base 13 at all, and thus 54 would be a red herring in a search for the question.

From a purely mathematical point of view, the most interesting thing about 42 could be that it could be the third moment of the

Riemann zeta function

, the leading coefficient of

1

T

⁢

∫

0

T

|

ζ

⁢

(

1

2

+

i

⁢

t

)

|

6

⁢

𝑑

t

expanded in powers of

log

⁡

(

T

)

.

The third

primary pseudoperfect number

and the

product

of the first three members of Sylvester’s

sequence

, 42 is the second

sphenic number

, but doesn’t seem to be the first of any interesting

integer sequence

(at least any sequence with the keywords ”core” or ”nice” in the OEIS). It does belong to several other interesting sequences, however, that of the

Catalan numbers

to name just one more.

When the 42nd

Mersenne prime

was discovered in 2005, Weisstein jokingly remarked that the answer ”is somehow

contained

in the 7.8 million decimal digits of that” prime. Berggren, Borwein and Borwein (authors of

Pi: A Source Book

) have also looked for 42 in the

decimal expansions

of

π

and

1

π

, finding ”042” at the fifty billionth position in both.

References

1

J. P. Keating and N. C. Snaith, “Random Matrix Theory and

θ

⁢

(

1

2

+

i

⁢

t

)

”,

Commun. Math. Phys.

214

, (2000): 57 - 89

2

L. Berggren, J. Borwein, and P. Borwein,

Pi: A Source Book

, New York: Springer-Verlag (1997)

3

E. Weisstein, “

http://mathworld.wolfram.com/42.html

42”

Mathworld

Title

forty-two

Canonical name

Fortytwo

Date of creation

2013-03-22 16:28:37

Last modified on

2013-03-22 16:28:37

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

8

Author

CompositeFan (12809)

Entry type

Feature

Classification

msc 11A99

Generated on Fri Feb 9 15:25:08 2018 by

LaTeXML
four hundred ninety-six

four hundred ninety-six

The third

perfect number

,

four hundred ninety-six

(496) has been known since antiquity. With just one larger perfect number known to him, Euclid was able to prove that all even perfect numbers are the

product

of a

Mersenne prime

and the nearest smaller

power of two

. In the case of 496, these are 31 and 16.

As a counterexample, 496 disproves Thomas Greenwood’s conjecture that an even

triangular number

with a prime index is one less than a prime, since although 496 is the 31st triangular number, 497 is not a prime.

496 is an important number in physics, and specifically string theory. “The massless bosonic states in this theory consist of a

symmetric

rank two field, an anti-symmetric rank two field, a scalar field known as the dilaton and a set of 496 gauge fields filling up the

adjoint representation

of the gauge group

E

8

×

E

8

.” (Sen, 1998) This discovery of the importance of 496, by Michael Green and John Schwartz is credited with ushering in an era of important revelations in string theory.

References

1

D. Wells

The Dictionary of Curious and Interesting Numbers

Suffolk: Penguin Books (1987): 155

2

A. Sen “An

Introduction

to Non-perturbative String Theory”

http://arxiv.org/abs/hep-th/9802051v1

ArXiv preprint

Title

four hundred ninety-six

Canonical name

FourHundredNinetysix

Date of creation

2013-03-22 17:10:51

Last modified on

2013-03-22 17:10:51

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Feature

Classification

msc 11A99

Synonym

four hundred and ninety-six

Generated on Fri Feb 9 15:25:10 2018 by

LaTeXML
irrational to an irrational power can be rational

irrational to an irrational power can be rational

Let

A

=

2

2

.
If

A

is a

rational number

, then it has the required property.
If

A

is an

irrational number

, let

B

=

A

2

,
then

B

=

2

2

=

2

is a rational.
Hence an irrational number to an irrational power can be a rational number.
(In fact, it follows from the

http://

planetmath

.org/node/3952

Gelfond-Schneider Theorem

that

A

is

transcendental

, and therefore irrational.)

Title

irrational to an irrational power can be rational

Canonical name

IrrationalToAnIrrationalPowerCanBeRational

Date of creation

2013-03-22 13:42:35

Last modified on

2013-03-22 13:42:35

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

10

Author

Koro (127)

Entry type

Result

Classification

msc 11A99

Related topic

TechniquesInMathematicalProofs

Generated on Fri Feb 9 15:25:11 2018 by

LaTeXML
multiplication operator

multiplication operator

A

multiplication operator

is an

operator

signifying

multiplication

among two or more operands. Common multiplication operators are

×

,

⋅

and

∗

as well as the

tacit multiplication operator

. The iterated multiplication operator is the Greek capital letter

Π

. Most of these can be used in

Polish notation

or

reverse Polish notation

just as easily as in standard

infix notation

. For example:

2

⋅

3

⋅

7

or

2

3

7

×

.

Most

computer

programming languages (such as

FORTRAN

and C++ (

http://planetmath.org/C

)) use the asterisk (

∗

), which is available as Shift-6 on almost all American typewriters and computer keyboards. For

arithmetic

computations

by hand on paper, the cross (

×

) is prefered and this is the symbol that appears on the multiplication key of most

calculators

. The tacit multiplication operator is mostly used in

algebra

to multiply named

variables

or

constants

, or a single

literal

by single or

multiple

named variables or constants.

Title

multiplication operator

Canonical

name

MultiplicationOperator

Date of creation

2013-03-22 16:46:32

Last modified on

2013-03-22 16:46:32

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A99

Generated on Fri Feb 9 15:25:13 2018 by

LaTeXML
number of the beast

number of the beast

In the Book of Revelation of the

Holy Bible

, verse 13:18 in the King James Version reads “Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.” The

translation

by Ulrich Zwingli gives the number of the beast as 616. Theologians have given various different explanations for these numbers.

If 666 appears more mathematically interesting than 616, it could be because professional mathematicians and amateurs alike have studied it more. Thus it has been noticed that the sum of the squares of the first seven primes is 666, that 666 is a doubly

triangular number

, that the 666th member of the

Padovan sequence

is a prime, etc. 666 would be a

pandigital number

in Roman numerals if it weren’t for the fact that it lacks the ’digit’ M. In base 10, the number 666 is a

Smith number

and also a

repdigit

(and a palidromic number as a trivial consequence of that).

Whereas 666 points to a prime in the Padovan sequence, 616 is itself a member of that

sequence

. Other than that, there doesn’t seem to be much about 616 that is interesting. The T-shirt that jokes “333: I’m Only Half Evil” would be a much less funny 308.

Title

number of the beast

Canonical name

NumberOfTheBeast

Date of creation

2013-03-22 16:29:33

Last modified on

2013-03-22 16:29:33

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A99

Synonym

beast number

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

LaTeXML
one hundred sixty-three

one hundred sixty-three

Of Martin Gardner’s April Fool’s hoaxes, perhaps the most famous comes from the April 1975 issue of

Scientific American

, in which he claimed that

Srinivasa Ramanujan

had proven that

e

π

⁢

163

is exactly equal to an

integer

. The truth is that it is not, but it comes surprisingly close, being approximately .0000000000007499274028018143 short of the nearest integer.

One hundred sixty-three

also appears in an approximation of

π

, namely,

2

9

163

≈

π

,

but this is barely correct to three decimal digits.

There are other qualities of 163 that are somewhat more exact, such as the fact that

∑

i

=

0

4

(

8

i

)

=

163

.

Kurt Heegner

proved that

n

=

163

is the largest value for which the

imaginary quadratic field

ℚ

⁢

(

-

n

)

has a

unique factorization

(thus 163 is the largest

Heegner number

).

Among the real integers, 163 is a

prime number

. As

163

+

0

⁢

i

, it is also a prime on the

complex plane

, that is, a

Gaussian prime

.

163 is the eighth prime that is not a

Chen prime

. Nor is it a

palindromic prime

in any base from binary to base 161 (hence it’s a

strictly non-palindromic number

).

Title

one hundred sixty-three

Canonical name

OneHundredSixtythree

Date of creation

2013-03-22 16:54:16

Last modified on

2013-03-22 16:54:16

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Feature

Classification

msc 11A99

Synonym

one hundred and sixty-three

Generated on Fri Feb 9 15:25:17 2018 by

LaTeXML
summed numerator and summed denominator

summed numerator and summed denominator

If

a

1

b

1

,

…

,

a

n

b

n

are any real

fractions

with

positive

denominators and

m

:=

min

⁡

{

a

1

b

1

,

…

,

a

n

b

n

}

,

M

:=

max

⁡

{

a

1

b

1

,

…

,

a

n

b

n

}

are the least and the greatest (

http://planetmath.org/MinimalAndMaximalNumber

) of the fractions, then

m

≦

a

1

+

…

+

a

n

b

1

+

…

+

b

n

≦

M

.

(1)

The equality signs are valid if and only if all fractions are equal; in this case one has

a

1

b

1

=

…

=

a

n

b

n

=

a

1

+

…

+

a

n

b

1

+

…

+

b

n

.

Proof.

Set

q

1

:=

a

1

b

1

,  …,

q

n

:=

a

n

b

n

.  Then we have

a

1

+

…

+

a

n

=

b

1

⁢

q

1

+

…

+

b

n

⁢

q

n

,  which apparently has the

lower bound

(

b

1

+

⋯

+

b

n

)

⁢

m

and the

upper bound

(

b

1

+

…

+

b

n

)

⁢

M

.  Dividing the three last expressions by the sum

b

1

+

…

+

b

n

yields the asserted double

inequality

(1).

Remark.

Cf. also the

mediant

.

Title

summed numerator and summed denominator

Canonical name

SummedNumeratorAndSummedDenominator

Date of creation

2013-10-11 15:35:42

Last modified on

2013-10-11 15:35:42

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A99

Related topic

InequalityForRealNumbers

Generated on Fri Feb 9 15:25:18 2018 by

LaTeXML
Suranyi’s theorem

Suranyi’s theorem

Suranyi’s

theorem

states that every integer

k

can be expressed as the following sum:

k

=

±

1

2

±

2

2

±

⋯

±

m

2

for some

m

∈

ℤ

+

.

We prove this by

induction

, taking the first four whole numbers as our cases:

0

=

1

2

+

2

2

-

3

2

+

4

2

-

5

2

-

6

2

+

7

2

1

=

1

2

2

=

-

1

2

-

2

2

-

3

2

+

4

2

3

=

-

1

2

+

2

2

Now it suffices to prove that if the theorem is true for

k

then
it is also true for

k

+

4

.

As

(

m

+

1

)

2

-

(

m

+

2

)

2

-

(

m

+

3

)

2

+

(

m

+

4

)

2

=

4

it’s simple to finish the proof:

if

k

=

±

1

2

±

⋯

±

m

2

then

(

k

+

4

)

=

±

1

2

±

⋯

±

m

2

+

(

m

+

1

)

2

-

(

m

+

2

)

2

-

(

m

+

3

)

2

+

(

m

+

4

)

2

and we are done.

Title

Suranyi’s theorem

Canonical name

SuranyisTheorem

Date of creation

2013-03-22 13:43:00

Last modified on

2013-03-22 13:43:00

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 11A99

Generated on Fri Feb 9 15:25:20 2018 by

LaTeXML
tacit multiplication operator

tacit multiplication operator

A

tacit multiplication operator

is a

multiplication operator

not actually written in a formula but usually understood from the

context

, at least by humans. For example, the

expression

2

⁢

π

⁢

r

is understood to mean

2

×

π

×

r

. Most advanced

computer algebra systems

(such as

Mathematica

) and even the Google calculator are capable of understanding the tacit multiplication operator provided that it does not clash in meaning with another possible command.

Title

tacit multiplication operator

Canonical

name

TacitMultiplicationOperator

Date of creation

2013-03-22 16:29:36

Last modified on

2013-03-22 16:29:36

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A99

Generated on Fri Feb 9 15:25:22 2018 by

LaTeXML