aarjaneiro/planetmath huggingface dataset:
conjecture on fractions with odd denominators

conjecture on fractions with odd denominators

Egyptian fractions

raise many

open problems

; this is one of the most famous of them.

Suppose we wish to write

fractions

as sums of distinct

unit fractions

with odd denominators

. Obviously, every such sum will have a

reduced

representation with an odd denominator.

For

instance

, the

greedy algorithm

applied to

2

7

gives

1

4

+

1

28

, but we may also write

2

7

as

1

7

+

1

9

+

1

35

+

1

315

.

It is known that we can we represent

every

rational number

with odd denominator as a sum of distinct unit fractions with odd denominators.

However it is not known whether the greedy algorithm (

http://

planetmath

.org/AnyRationalNumberIsASumOfUnitFractions

) works when limited to odd denominators.

Conjecture 1

.

For any fraction

0

≤

a

2

⁢

k

+

1

<

1

with odd denominator, if we repeatedly subtract the largest unit fraction with odd denominator that is smaller than our fraction, we will

eventually

reach 0.

Title

conjecture on fractions with odd denominators

Canonical name

ConjectureOnFractionsWithOddDenominators

Date of creation

2013-03-22 12:48:34

Last modified on

2013-03-22 12:48:34

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Conjecture

Classification

msc 11D68

Classification

msc 11A67

Related topic

SierpinskiErdosEgyptianFractionConjecture

Generated on Fri Feb 9 15:32:20 2018 by

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Sierpinski Erdős egyptian fraction conjecture

Sierpinski Erdős egyptian fraction conjecture

Erdős and Sierpinski conjectured that for any

integer

n

>

3

there exist

positive integers

a

,

b

,

c

so that:

5

n

=

1

a

+

1

b

+

1

c

Title

Sierpinski Erdős

egyptian fraction

conjecture

Canonical name

SierpinskiErdHosEgyptianFractionConjecture

Date of creation

2013-03-22 13:43:12

Last modified on

2013-03-22 13:43:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Conjecture

Classification

msc 11D68

Classification

msc 11A67

Synonym

Sierpiński Erdős egyptian fraction conjecture

Related topic

UnitFraction

Related topic

AnyRationalNumberWithOddDenominatorIsASumOfUnitFractionsWithOddDenominators

Generated on Fri Feb 9 15:32:22 2018 by

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Markov number

Markov number

A

Markov number

is an

integer

x

,

y

or

z

that fits in the

Diophantine equation

x

2

+

y

2

+

z

2

=

3

⁢

x

⁢

y

⁢

z

and gives a

Lagrange number

L

x

=

9

-

4

x

2

(or

y

or

z

as the case may be).

The solutions, (1, 1, 1), (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), (1, 13, 34), (1, 34, 89), (2, 29, 169), (5, 13, 194), (1, 89, 233), etc., can be put in a binary graph tree. Thus arranged, the numbers on 1’s branch are

Fibonacci numbers

with odd index, and the numbers on 2’s branch are

Pell numbers

with odd index.

Georg Frobenius

proved that, with the exception of the smallest Markov triple, the numbers in a Markov triple are

pairwise coprime

. He also proved that an odd Markov number

x

≡

1

mod

4

(or

y

or

z

) and an even Markov number

x

≡

2

mod

8

. Ying Zhang used this to prove that even Markov numbers satisfy the sharper

congruence

x

≡

2

mod

32

, which he calls the best possible since the first two even Markov numbers are 2 and 34.

References

1

Ying Zhang, “Congruence and Uniqueness of Certain Markov Numbers”

Acta Arithmetica

128

3 (2007): 297

Title

Markov number

Canonical name

MarkovNumber

Date of creation

2013-03-22 15:46:19

Last modified on

2013-03-22 15:46:19

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

10

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11D72

Classification

msc 11J06

Synonym

Markoff number

Generated on Fri Feb 9 15:32:24 2018 by

LaTeXML
Chinese remainder theorem

Chinese remainder theorem

Suppose we have a set of

n

congruences

of the form

x

≡

a

1

(

mod

p

1

)

x

≡

a

2

(

mod

p

2

)

⋮

x

≡

a

n

(

mod

p

n

)

where

p

1

,

p

2

,

…

,

p

n

are relatively prime. Let

P

=

∏

i

=

1

n

p

i

and, for all

i

∈

ℕ

(

1

≤

i

≤

n

), let

y

i

be an

integer

that

satisfies

y

i

⋅

P

p

i

≡

1

(

mod

p

i

)

Then one solution of these congruences is

x

0

=

∑

i

=

1

n

a

i

⁢

y

i

⋅

P

p

i

Any

x

∈

ℤ

satisfies the set of congruences if and only if it satisfies

x

≡

x

0

(

mod

P

)

The

Chinese remainder theorem

originated in the book

“Sun Zi Suan Jing”

, or Sun Tzu’s

Arithmetic

Classic, by the Chinese mathematician Sun Zi, or Sun Tzu, who also wrote “Sun Zi Bing Fa”, or Sun Tzu’s The Art of War. The

theorem

is said to have been used to count the size of the ancient Chinese armies (i.e., the soldiers would split into groups of 3, then 5, then 7, etc, and the “leftover” soldiers from each grouping would be counted).

Title

Chinese remainder theorem

Canonical name

ChineseRemainderTheorem

Date of creation

2013-03-22 11:58:58

Last modified on

2013-03-22 11:58:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 11D79

Related topic

ChineseRemainderTheoremInTermsOfDivisorTheory

Related topic

GodelsBetaFunction

Generated on Fri Feb 9 15:32:26 2018 by

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Marshall Hall’s conjecture

Marshall Hall’s conjecture

Conjecture

. (Marshall Hall, Jr.). With the exception of

n

2

being a perfect sixth power, for any

positive integer

n

, the

inequality

|

n

2

-

m

3

|

>

C

⁢

m

, (with

m

also being a positive integer and

C

being a number less than 1 that nears 1 as

n

tends to

infinity

) always holds.

The reason for the exception of perfect sixth powers (those cases of

n

for which there is a solution to

n

2

=

h

6

in

integers

) is a simple

consequence

of associativity: if

n

2

=

h

6

, then

h

6

=

h

2

⁢

h

2

⁢

h

2

=

h

3

⁢

h

3

. Then

m

=

h

and

n

2

-

m

3

=

0

. For example,

8

2

-

4

3

=

0

.

For small

n

,

C

can’t be exactly 1. For example,

3

2

-

2

3

=

1

, and

2

>

1

. But even among the smaller numbers, the conjecture generally holds even with

C

=

1

. After

n

=

3

, the next

counterexample

(that is not a perfect sixth power) to

C

=

1

is

n

=

378661

, with the corresponding

m

=

5234

producing a

difference

of just 17. A078933 in Sloane’s OEIS lists smaller values of

m

with cubes being at a distance from the nearest square that is less than

m

. Noam Elkies has found some fairly large counterexamples to setting

C

=

1

, such as

n

=

447884928428402042307918

and

m

=

5853886516781223

, the difference between the square of the former and the cube of the latter being a relatively small 1641843.

References

1

R. K. Guy,

Unsolved Problems in

Number Theory

New York: Springer-Verlag 2004: D9

Title

Marshall Hall’s conjecture

Canonical

name

MarshallHallsConjecture

Date of creation

2013-03-22 18:15:36

Last modified on

2013-03-22 18:15:36

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11D79

Synonym

Marshall Hall conjecture

Related topic

PerfectPower

Generated on Fri Feb 9 15:32:31 2018 by

LaTeXML
Agoh-Giuga conjecture

Agoh-Giuga conjecture

In 1950, Giuseppe Giuga conjectured that if and only if an

integer

p

is prime then it will satisfy the

congruence

∑

i

=

1

p

-

1

i

p

-

1

≡

-

1

mod

p

.

This is sometimes called the

Giuga conjecture

. Takashi Agoh rephrased the

conjecture

as

p

⁢

B

p

-

1

≡

-

1

mod

p

, where

B

is a

Bernoulli number

; this is called the

Agoh-Giuga conjecture

. In 2003 Simon Plouffe performed an

exhaustive search

for a

counterexample

below 50000 but came up empty.

Title

Agoh-Giuga conjecture

Canonical

name

AgohGiugaConjecture

Date of creation

2013-03-22 16:17:55

Last modified on

2013-03-22 16:17:55

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11D85

Synonym

Giuga conjecture

Synonym

Giuga’s conjecture

Synonym

Agoh conjecture

Synonym

Agoh’s conjecture

Generated on Fri Feb 9 15:32:33 2018 by

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Giuga number

Giuga number

A

Giuga number

is a

squarefree

composite number

n

such that each

prime factor

p

i

|

(

n

p

i

-

1

)

. For these numbers it then follows that

n

⁢

B

ϕ

⁢

(

n

)

≡

-

1

mod

n

, (where

B

x

is a

Bernoulli number

).

The first few Giuga numbers are 30, 858, 1722, 66198, 2214408306, 24423128562 (listed in

sequence

A007850 of Sloane’s OEIS).

All known Giuga numbers are even and have at least three factors. An odd Giuga number would have to have at least twelve factors.

Title

Giuga number

Canonical name

GiugaNumber

Date of creation

2013-03-22 15:50:22

Last modified on

2013-03-22 15:50:22

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

7

Author

Mravinci (12996)

Entry type

Definition

Classification

msc 11D85

Related topic

PrimaryPseudoperfectNumber

Generated on Fri Feb 9 15:32:35 2018 by

LaTeXML
oblong number

oblong number

An

integer

of the form

n

⋅

(

n

+

1

)

or

n

2

+

n

. The

partial sum

∑

i

=

0

n

2

⁢

i

works, too. From this it follows that one can also multiply the

n

th

triangular number

. Plenty of other relations to various

polygonal numbers

can be obtained.

The first few terms are 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, and are listed in A002378 of Sloane’s OEIS.

Title

oblong number

Canonical name

OblongNumber

Date of creation

2013-03-22 15:50:19

Last modified on

2013-03-22 15:50:19

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11D85

Synonym

heteromecic number

Synonym

pronic number

Generated on Fri Feb 9 15:32:36 2018 by

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polygonal number

polygonal number

A

polygonal number

, or

figurate number

, is any value of the

function

P

d

⁢

(

n

)

=

(

d

-

2

)

⁢

n

2

+

(

4

-

d

)

⁢

n

2

for

integers

n

≥

0

and

d

≥

3

.
A “generalized polygonal number”
is any value of

P

d

⁢

(

n

)

for some integer

d

≥

3

and any

n

∈

ℤ

.
For fixed

d

,

P

d

⁢

(

n

)

is called a

d

-gonal or

d

-polygonal number.
For

d

=

3

,

4

,

5

,

…

, we speak of a

triangular number

, a

square
number

or a square, a

pentagonal number

, and so on.

An

equivalent

definition of

P

d

, by

induction

on

n

, is:

P

d

⁢

(

0

)

=

0

P

d

⁢

(

n

)

=

P

d

⁢

(

n

-

1

)

+

(

d

-

2

)

⁢

(

n

-

1

)

+

1

for all

⁢

n

≥

1

P

d

⁢

(

n

-

1

)

=

P

d

⁢

(

n

)

+

(

d

-

2

)

⁢

(

1

-

n

)

-

1

for all

⁢

n

<

0

.

From these equations, we can deduce that all generalized polygonal
numbers are

nonnegative integers

.
The first two

formulas

show that

P

d

⁢

(

n

)

points can be arranged in a
set of

n

nested

d

-gons, as in this diagram of

P

3

⁢

(

5

)

=

15

and

P

5

⁢

(

5

)

=

35

.

Polygonal numbers were studied somewhat by the ancients, as far
back as the Pythagoreans, but nowadays their interest
is mostly historical, in connection with this famous result:

Theorem

:

For any

d

≥

3

, any integer

n

≥

0

is the
sum of some

d

d

-gonal numbers.

In other words, any nonnegative integer is a sum of three
triangular numbers, four squares, five pentagonal numbers,
and so on.
Fermat made this remarkable statement in a letter to Mersenne.
Regrettably, he never revealed the

argument

or proof that he
had in mind. More than a century passed before Lagrange proved
the easiest case: Lagrange’s four-square theorem. The
case

d

=

3

was demonstrated by Gauss around 1797, and the
general case by Cauchy in 1813.

Title

polygonal number

Canonical name

PolygonalNumber

Date of creation

2013-03-22 13:55:38

Last modified on

2013-03-22 13:55:38

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11D85

Classification

msc 11D09

Synonym

figurate number

Defines

pentagonal number

Generated on Fri Feb 9 15:32:39 2018 by

LaTeXML
primary pseudoperfect number

primary pseudoperfect number

Given an

integer

n

with

ω

⁢

(

n

)

distinct prime factors

p

i

(where

ω

is

number of distinct prime factors function

), if the equality

1

n

+

∑

i

=

1

ω

⁢

(

n

)

1

p

i

=

1

holds true, then

n

is a

primary pseudoperfect number

. Equivalently,

1

+

∑

i

=

1

ω

⁢

(

n

)

n

p

i

=

n

if

n

is a primary pseudoperfect number.

The first few primary pseudoperfect numbers are 2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086, the first four of these being each one less than the first four terms of Sylvester’s

sequence

; these are listed in A054377 of Sloane’s OEIS. Presently it’s not known whether there are any odd primary pseudoperfect numbers.

Title

primary pseudoperfect number

Canonical name

PrimaryPseudoperfectNumber

Date of creation

2013-03-22 16:17:40

Last modified on

2013-03-22 16:17:40

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

6

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11D85

Related topic

GiugaNumber

Generated on Fri Feb 9 15:32:40 2018 by

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primitive semiperfect number

primitive semiperfect number

Parse the

sequence

of

semiperfect numbers

in

ascending order

, striking out any numbers that are

divisible

by any smaller number in the sequence. A number that remains after this process is then called a

primitive semiperfect number

or

primitive pseudoperfect number

. Put another way, none of the

divisors

of a primitive semiperfect number are themselves semiperfect. The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, listed in A6036 of Sloane’s OEIS.

The list of the largest prime factors of each of the primitive semiperfect numbers gives the list of prime numbers almost in order but with many duplicates interspersed.

The smallest odd primitive semiperfect number is 945, which is also the smallest odd

abundant number

.

Title

primitive semiperfect number

Canonical name

PrimitiveSemiperfectNumber

Date of creation

2013-03-22 16:19:00

Last modified on

2013-03-22 16:19:00

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

5

Author

Mravinci (12996)

Entry type

Definition

Classification

msc 11D85

Synonym

primitive pseudoperfect number

Synonym

irreducible semiperfect number

Generated on Fri Feb 9 15:32:42 2018 by

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semiperfect number

semiperfect number

Given an

integer

n

and the subsets of its

proper divisors

d

i

|

n

and

d

i

<

n

(thus

0

<

i

<

τ

⁢

(

n

)

where

τ

is the

divisor function

), is there at least one subset whose elements add up to

n

? If yes, then

n

is a

semiperfect number

or

pseudoperfect number

.

Since the

complete set

of proper divisors is also technically considered a subset, then a fully

perfect number

is also a semiperfect number. Perhaps just as obviously, no

deficient number

can be semiperfect; thus all semiperfect numbers are either

abundant numbers

or perfect numbers.

If the

abundance

a

⁢

(

n

)

happens to be a

divisor

of

n

, then the divisor subset that excludes

a

⁢

(

n

)

is the obvious choice, but some semiperfect numbers are so in more than one way: 12 for example can be expressed as 1 + 2 + 3 + 6 but also as 2 + 4 + 6.

Just as a

multiple

of an abundant number is another abundant number, so is the multiple of a semiperfect number another semiperfect number.

The first few semiperfect numbers that are not multiples of perfect numbers are 20, 40, 80, 88. A005835 of Sloane’s OEIS lists all the semiperfect numbers less than 265 and provides a simple means of reckoning them, by counting the number of

partitions

of

n

into distinct divisors and culling those that have more than 1.

All

primary pseudoperfect numbers

(except 2) are also semiperfect. An abundant number that is not semiperfect is a

weird number

.

Title

semiperfect number

Canonical name

SemiperfectNumber

Date of creation

2013-03-22 16:18:33

Last modified on

2013-03-22 16:18:33

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11D85

Synonym

pseudoperfect number

Generated on Fri Feb 9 15:32:44 2018 by

LaTeXML
weird number

weird number

Given an

abundant number

n

, if no subset of its

divisors

d

1

,

…

,

d

τ

⁢

(

n

)

-

1

(where

τ

is the

divisor function

) can be selected that adds up to

n

(that is,

n

is

not

a

semiperfect number

), then

n

is said to be a

weird number

.

The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, (listed in A006037 of Sloane’s OEIS). All the known weird numbers are even. In 1972, Benkoski wondered if there are any odd weird numbers; to this day

10

17

is an accepted

lower bound

. Despite this question of parity, it has been proven that there are infinitely many weird numbers and that they have

positive

Schnirelmann density

.

Even so, weird numbers are rarer than semiperfect numbers; twice a weird number is usually a semiperfect number, which makes all subsequent

multiples

of a weird number also semiperfect.

0.1

Trivia

The band Boards of Canada records for the label Music70. Track 9 of their album

Geogaddi

is titled ”The Smallest Weird Number.”

References

1

S. Benkoski, ”Are All Weird Numbers Even?”,

Amer. Math. Monthly

79

(1972), 774.

2

Wikipedia,

http://en.wikipedia.org/wiki/Weird_number

Weird number

Title

weird number

Canonical name

WeirdNumber

Date of creation

2013-03-22 16:18:54

Last modified on

2013-03-22 16:18:54

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

6

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11D85

Generated on Fri Feb 9 15:32:46 2018 by

LaTeXML
Diophantine equation

Diophantine equation

A

Diophantine equation

is an equation between

polynomials

in finitely many

variables

over the

integers

. Usually one is interested in integral solutions.

Generally, solving a Diophantine equation is not as straightforward as solving a

similar

equation in the real numbers. For example, consider this equation:

x

4

+

y

4

=

z

4

It is easy to find real numbers

x

,

y

,

z

that

satisfy

this equation: pick any arbitrary

x

and

y

, and you can compute a

z

from them. But if we require that

x

,

y

,

z

all be integers, it is no longer

obvious

at all how to find solutions. Even though raising an integer to an integer power yields another integer, the reverse is not true in general.

As it turns out, of course, there are no solutions to the above Diophantine equation: it is a case of Fermat’s last

theorem

.

At the Second

International Congress of Mathematicians

in 1900,

David Hilbert

presented several unsolved problems in mathematics that he believed held special importance. Hilbert’s tenth problem was to find a general procedure for determining if Diophantine equations have solutions:

“Given a Diophantine equation with any number of unknowns and with

rational integer

coefficients: devise a process, which could determine by a finite number of

operations

whether the equation is solvable in rational integers.”

Note that this preceded the formal study of computing and Gödel’s

incompleteness theorem

, and it is unlikely that Hilbert had anticipated a

negative

solution — that is, a proof that no such algorithm is possible — but that turned out the be the case. Exponential Diophantine equations are similar to Diophantine equations, except that polynomials as well as integers are permitted as exponents. In the 1950s and 60s, Martin Davis,

Julia Robinson

, and Hilary Putnam showed that an algorithm to determine the solubility of all exponential Diophantine equations is impossible. Yuri Matiyasevich extended that work in 1970 by showing that there is no algorithm for determining whether an arbitrary Diophantine equation has integral solutions.

References:

Matiyasevich, Yuri V.,

Hilbert’s Tenth Problem

, MIT, 1993.

Title

Diophantine equation

Canonical

name

DiophantineEquation

Date of creation

2013-03-22 11:59:14

Last modified on

2013-03-22 11:59:14

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 11D99

Synonym

Diophantine

Related topic

Equation

Related topic

FromHilbertsTenthProblemToGodelsTrichotomy

Generated on Fri Feb 9 15:32:48 2018 by

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solution of 1/x+1/y=1/n

solution of

1

/

x

+

1

/

y

=

1

/

n

Theorem 1

.

Given an integer

n

, if there exist integers

x

and

y

such that

1

x

+

1

y

=

1

n

,

then one has

x

=

n

⁢

(

u

+

v

)

u

y

=

n

⁢

(

u

+

v

)

v

where

u

and

v

are integers such that

u

⁢

v

divides

n

.

Proof.

To begin, cross multiply to obtain

x

⁢

y

=

n

⁢

(

x

+

y

)

.

Since this involves setting a

product

equal to another
product, we can think in terms of factorization. To
clarify things, let us pull out a common factor and
write

x

=

k

⁢

v

and

y

=

k

⁢

u

, where

k

is the

greatest
common factor

and

u

is relatively prime to

v

. Then,
cancelling a common factor of

k

, our equation becomes
the following:

k

⁢

u

⁢

v

=

n

⁢

(

u

+

v

)

This is

equivalent

to

u

v

∣

n

(

u

+

v

)

Since

u

and

v

are relatively prime, it follows that

u

is
relatively prime to

u

+

v

and that

v

is relatively prime to

u

+

v

as well. Hence, we must have that

u

⁢

v

divides

n

,

Now we can obtain the general solution to the equation.
Write

n

=

m

⁢

u

⁢

v

with

u

and

v

relatively prime. Then,
substituting into our equation and cancelling a

u

and a

v

, we obtain

k

=

m

⁢

(

u

+

v

)

,

so the solution to the original equation is

x

=

m

⁢

v

⁢

(

u

+

v

)

y

=

m

⁢

u

⁢

(

u

+

v

)

Using the definition of

m

, this can be rewritten as

x

=

n

⁢

(

u

+

v

)

u

y

=

n

⁢

(

u

+

v

)

v

.

∎

Title

solution of

1

/

x

+

1

/

y

=

1

/

n

Canonical name

SolutionOf1x1y1n

Date of creation

2013-03-22 16:30:44

Last modified on

2013-03-22 16:30:44

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 11D99

Generated on Fri Feb 9 15:32:50 2018 by

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