aarjaneiro/planetmath huggingface dataset:
Higgs prime

Higgs prime

A

Higgs prime

is a

prime number

H

⁢

p

n

for which, given an

exponent

a

, it is the case that

ϕ

(

H

p

n

)

|

∏

i

=

1

n

-

1

H

p

i

a

,

(where

ϕ

⁢

(

x

)

is Euler’s

totient function

) and

H

⁢

p

n

>

H

⁢

p

n

-

1

.

For

a

=

2

, the first few Higgs primes are 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, etc., listed in A007459 of Sloane’s OEIS. So, for example, 13 is a Higgs prime because the square of the product of the smaller Higgs primes is 5336100, and divided by 12 this is 444675. But 17 is not a Higgs prime because the square of the product of the smaller primes is 901800900, which leaves a

remainder

of 4 when divided by 16.

From observation of the first few Higgs primes for squares through seventh powers, it would seem more compact to list those primes that are not Higgs primes. Observation further reveals that a

Fermat prime

2

2

n

+

1

can’t be a Higgs prime for the

a

th power if

a

<

2

n

.

It’s not known if there are infinitely many Higgs primes for any exponent

a

>

1

. The situation is quite different for

a

=

1

. There are only four of them: 2, 3, 7 and 43 (a

sequence

suspiciously similar to Sylvester’s sequence). In 1993, Burris and Lee found that about a fifth of the primes below a million are Higgs prime, and they concluded that even if the sequence of Higgs primes for squares is finite, “a computer enumeration is not feasible.”

References

1

S. Burris & S. Lee, “Tarski’s high school

identities

”,

Amer. Math. Monthly

100

(1993): 233

2

N. Sloane & S. Plouffe,

The Encyclopedia of

Integer Sequences

, New York: Academic Press (1995): M0660

Title

Higgs prime

Canonical name

HiggsPrime

Date of creation

2013-03-22 16:43:12

Last modified on

2013-03-22 16:43:12

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Synonym

Higgs’ prime

Synonym

Higgs’s prime

Synonym

Higg’s prime

Synonym

Higg prime

Generated on Fri Feb 9 15:16:28 2018 by

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Home prime

Home prime

The

home prime

of a number is found by repeated

concatenation

of its

prime factors

until a

prime number

is obtained. For example, the home prime of 9 is 311 since 9 = 3*3 giving 33 which is

composite

; 33 = 3*11 giving 311 which is a prime number. So, the home prime of n for n = 2, 3, 4 … are 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 733, 11, 233 … (

Sequence

A037274 of OEIS). The number of steps needed to obtain the home prime of n for n = 2, 3, 4 … are 0, 0, 2, 0, 1, 0, 13, 2, 4, 0, 1, 0, 5, 4, 4, 0, 1 … (Sequence A037273 of OEIS). It is believed that every number has a homeprime. The smallest number whose homeprime is not known is 49.
It is checked upto 109-th step and all

resultant

concatenations are composite. Similarly, the first few numbers whose homeprime is not known are 77, 49, 146, 246, 312, 320 … At least 50 concatenations of all these numbers are verified.

Title

Home prime

Canonical name

HomePrime

Date of creation

2013-03-22 19:20:36

Last modified on

2013-03-22 19:20:36

Owner

Kausthub (26471)

Last modified by

Kausthub (26471)

Numerical id

4

Author

Kausthub (26471)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:16:32 2018 by

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integer factorization

integer factorization

Given an

integer

n

, its

integer factorization

(or

prime factorization

) consists of the primes

p

i

which multiplied together give

n

as a result. To put it algebraically,

n

=

∏

i

=

1

ω

⁢

(

n

)

p

i

a

i

,

with each

p

i

distinct, all

a

i

>

0

but not necessarily distinct, and

ω

⁢

(

n

)

being the value of the

number of distinct prime factors function

. Theoretically, an integer is a product of all the

prime numbers

,

n

=

∏

i

=

1

∞

p

i

a

i

,

with many

a

i

=

0

.

For example, the factorization of 32851 is

7

×

13

×

19

×

19

, more usually expressed as

7

×

13

×

19

2

. Because of the commutative property of

multiplication

, it does not matter in what order the

prime factors

are stated in, but it is customary to give them in ascending order (

http://planetmath.org/AscendingOrder

), and to group them together by the use of exponents.

The factorization of a

positive integer

is unique (this is the

fundamental theorem of arithmetic

). For a

negative number

n

<

0

one could take the factorization of

|

n

|

and randomly give

negative

signs to one (or any

odd number

) of the prime factors. Alternatively, the factorization can be given as

-

1

⋅

p

1

a

1

⋅

…

(this is what

Mathematica

opts for).

The term “factorization” is often used to refer to the actual process of determining the prime factors. There are several

algorithms

to choose from, with

trial division

being the simplest to implement.

Title

integer factorization

Canonical

name

IntegerFactorization

Date of creation

2013-03-22 16:39:09

Last modified on

2013-03-22 16:39:09

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Synonym

prime factorization

Generated on Fri Feb 9 15:16:34 2018 by

LaTeXML
Legendre’s conjecture

Legendre’s conjecture

(

Adrien-Marie Legendre

) There is always a

prime number

between a

square number

and the next. To put it algebraically, given an

integer

n

>

0

, there is always a prime

p

such that

n

2

<

p

<

(

n

+

1

)

2

. Put yet another way,

(

π

⁢

(

(

n

+

1

)

2

)

-

π

⁢

(

n

2

)

)

>

0

, where

π

⁢

(

x

)

is the

prime counting function

.

This

conjecture

was considered unprovable when it was listed in Landau’s problems in 1912. Almost a

hundred

years later, the conjecture remains unproven even as similar conjectures (such as Bertrand’s postulate) have been proven.

But progress has been made.

Chen Jingrun

proved a slightly weaker version of the conjecture: there is either a prime

n

2

<

p

<

(

n

+

1

)

2

or a

semiprime

n

2

<

p

⁢

q

<

(

n

+

1

)

2

(where

q

is a prime unequal to

p

). Thanks to

computers

, brute force searches have shown that the conjecture holds true as high as

n

=

10

5

.

Title

Legendre’s conjecture

Canonical

name

LegendresConjecture

Date of creation

2013-03-22 16:38:17

Last modified on

2013-03-22 16:38:17

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11A41

Related topic

BrocardsConjecture

Generated on Fri Feb 9 15:16:36 2018 by

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linear congruence

linear congruence

The

linear congruence

a

⁢

x

≡

b

(

mod

m

)

,

where

a

,

b

and

m

are known

integers

and

gcd

⁡

(

a

,

m

)

=

1

,  has exactly one solution

x

in

ℤ

, when numbers

congruent

to each other are not regarded as different.  The solution can be obtained as

x

=

a

φ

⁢

(

m

)

-

1

⁢

b

,

where

φ

means Euler’s phi-function.

Solving the linear congruence also gives the solution of the

Diophantine equation

a

⁢

x

-

m

⁢

y

=

b

,

and conversely.  If

x

=

x

0

,

y

=

y

0

is a solution of this equation, then the general solution is

{

x

=

x

0

+

k

⁢

m

,

y

=

y

0

+

k

⁢

a

,

where

k

=

0

,

±

1

,

±

2

, …

Title

linear congruence

Canonical name

LinearCongruence

Date of creation

2013-03-22 14:18:15

Last modified on

2013-03-22 14:18:15

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

15

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 11A41

Synonym

first degree congruence

Related topic

QuadraticCongruence

Related topic

SolvingLinearDiophantineEquation

Related topic

GodelsBetaFunction

Related topic

ConditionalCongruences

Generated on Fri Feb 9 15:16:37 2018 by

LaTeXML
lucky number

lucky number

A

lucky number

is an

integer

that remains after a sieving process

similar

to a sieving process for

prime numbers

. The first few lucky numbers are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, etc., listed in A000959 of Sloane’s OEIS. There are infinitely many lucky numbers. These numbers share some properties with prime numbers, mostly in regards to distribution. Stanisław Ulam (

http://planetmath.org/StanislawUlam

) was the first to study these numbers.

The sieve process for the lucky numbers begins with a list of odd

positive integers

from 1 to whatever limit one wishes (let’s say 49). We circle 1 and 3 and cross out every third number that remains (counting from the beginning):

1

,

3

,

5̸

,

7

,

9

,

11

,

13

,

15

,

17

,

19

,

…

Then we circle the number next to the one that we last circled and cross out every

x

th term as indicated by the number we just circled, in this case, 7, starting the count from the beginning but not

counting numbers

that have already been struck out:

1

,

3

,

5̸

,

7

,

9

,

11

,

13

,

15

,

17

,

19

,

…

This step is repeated until every number in our list has been either circled or crossed out. The numbers that remain are “lucky” because they survived the process without ever being struck.

The “lucky number

theorem

” is almost the same as the

prime number theorem

.

Title

lucky number

Canonical name

LuckyNumber

Date of creation

2013-03-22 16:55:28

Last modified on

2013-03-22 16:55:28

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:16:39 2018 by

LaTeXML
lucky prime

lucky prime

A

lucky prime

is a

lucky number

that is also a

prime number

. So, in a sense, such numbers are doubly “lucky” because they’ve survived two different sieves. The first few are 3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163 (

http://planetmath.org/OneHundredSixtyThree

), 193, etc., listed in A031157 of Sloane’s OEIS. It’s known that there are infinitely many primes and it’s known that there are infinitely many lucky numbers, but it’s not known if there are infinitely many lucky primes.

Title

lucky prime

Canonical

name

LuckyPrime

Date of creation

2013-03-22 16:55:31

Last modified on

2013-03-22 16:55:31

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:16:41 2018 by

LaTeXML
Mangoldt summatory function

Mangoldt summatory function

A

number theoretic function

used in the study of

prime numbers

; specifically it was used in the proof of the

prime number theorem

.

It is defined thus:

ψ

⁢

(

x

)

=

∑

r

≤

x

Λ

⁢

(

r

)

where

Λ

is the

Mangoldt function

.

The

Mangoldt summatory function

is valid for all

positive

real x.

Note that we do not have to worry that the

inequality

above is ambiguous, because

Λ

⁢

(

x

)

is only non-zero for natural

x

. So no matter whether we take it to mean r is real,

integer

or natural, the result is the same because we just get a lot of zeros added to our answer.

The prime number theorem, which states:

π

⁢

(

x

)

∼

x

ln

⁡

(

x

)

where

π

⁢

(

x

)

is the

prime counting function

, is

equivalent

to the statement that:

ψ

⁢

(

x

)

∼

x

We can also define a “smoothing

function

” for the

summatory function

, defined as:

ψ

1

⁢

(

x

)

=

∫

0

x

ψ

⁢

(

t

)

⁢

𝑑

t

and then the prime number theorem is also equivalent to:

ψ

1

⁢

(

x

)

∼

1

2

⁢

x

2

which turns out to be easier to work with than the original form.

Title

Mangoldt summatory function

Canonical

name

MangoldtSummatoryFunction

Date of creation

2013-03-22 13:27:16

Last modified on

2013-03-22 13:27:16

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11A41

Synonym

von Mangoldt summatory function

Related topic

ChebyshevFunctions

Generated on Fri Feb 9 15:16:44 2018 by

LaTeXML
Mangoldt summatory function is O⁢(x)

Mangoldt summatory function is

O

⁢

(

x

)

Theorem 1

ψ

⁢

(

x

)

=

O

⁢

(

x

)

, in other ,

ψ

⁢

(

x

)

x

is

bounded

.

Proof.

ψ

⁢

(

x

)

=

∑

1

x

Λ

⁢

(

n

)

=

∑

p

⁢

prime

p

≤

x

⌊

log

p

⁡

x

⌋

⁢

ln

⁡

p

=

∑

p

⁢

prime

p

≤

x

⌊

ln

⁡

x

ln

⁡

p

⌋

⁢

ln

⁡

p

=

∑

p

⁢

prime

p

≤

x

⌊

ln

⁡

x

ln

⁡

p

⌋

⁢

ln

⁡

p

+

∑

p

⁢

prime

x

<

p

≤

x

ln

⁡

p

since

1

≤

ln

⁡

x

ln

⁡

p

<

2

if

p

>

x

.
Continuing, we have

∑

p

⁢

prime

p

≤

x

⌊

ln

⁡

x

ln

⁡

p

⌋

⁢

ln

⁡

p

+

∑

p

⁢

prime

x

<

p

≤

x

ln

⁡

p

≤

x

⁢

ln

⁡

x

+

π

⁢

(

x

)

⁢

ln

⁡

x

≤

x

⁢

ln

⁡

x

+

8

⁢

x

⁢

ln

⁡

2

=

O

⁢

(

x

)

Note that

π

⁢

(

x

)

⁢

ln

⁡

x

≤

8

⁢

x

⁢

ln

⁡

2

by Chebyshev’s bounds on

π

⁢

(

x

)

(

http://planetmath.org/BoundsOnPin

).

Title

Mangoldt summatory function

is

O

⁢

(

x

)

Canonical name

MangoldtSummatoryFunctionIsOx

Date of creation

2013-03-22 17:42:59

Last modified on

2013-03-22 17:42:59

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11A41

Generated on Fri Feb 9 15:16:46 2018 by

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Mersenne numbers

Mersenne numbers

Numbers of the form

M

n

=

2

n

-

1

,

(

n

≥

1

)

are called

Mersenne numbers

after Father Marin Mersenne (1588 - 1648), a French monk who studied which of these numbers are actually prime. It can be easily shown that if

M

n

is prime then

n

is prime. Indeed,

2

a

⋅

b

-

1

with

a

,

b

>

1

factors:

2

a

⋅

b

-

1

=

(

2

a

-

1

)

⁢

(

2

a

⁢

(

b

-

1

)

+

2

a

⁢

(

b

-

2

)

+

…

+

2

a

+

1

)

.

If

M

n

is prime then we call it a

Mersenne prime

. Mersenne primes have a strong connection with

perfect numbers

.

The currently known Mersenne primes correspond to

n

= 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13,466,917 and the newly discovered

40

th

number

n

=

20996011

, and even newer

41

st

number

n

=

24036583

. The latest Mersenne primes (as of

2

/

5

/

2006

) are the

42

nd Mersenne number which corresponds to

n

=

25964951

(and which has more than

7.8

million digits) and the

43

rd Mersenne prime for

n

=

30402457

(the new prime is

9

,

152

,

052

digits long). For an updated list and a lot more information on how these numbers were discovered, you can check:

http://www.mersenne.org

www.mersenne.org.

It is conjectured that the density of Mersenne primes with

exponent

p

<

x

is of order

e

γ

log

⁡

2

⁢

log

⁡

log

⁡

x

where

γ

is Euler’s constant.

Title

Mersenne numbers

Canonical name

MersenneNumbers

Date of creation

2013-03-22 11:47:54

Last modified on

2013-03-22 11:47:54

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

19

Author

alozano (2414)

Entry type

Definition

Classification

msc 11A41

Classification

msc 11-02

Related topic

TwoSmallResultsMersenneNumbers

Defines

Mersenne prime

Generated on Fri Feb 9 15:16:48 2018 by

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Mersenne numbers, two small results on

Mersenne numbers, two small results on

This entry presents two simple results on

Mersenne numbers

1

1

In this entry, the Mersenne numbers are

indexed by

the primes.

, namely that any two Mersenne numbers are relatively prime and that any prime dividing a Mersenne number

M

p

is greater than

p

. We prove something slightly stronger for both these results:

Theorem

.

If

q

is a prime such that

q

∣

M

p

, then

p

∣

(

q

-

1

)

.

Proof.

By definition of

q

, we have

2

p

≡

1

(

mod

q

)

. Since

p

is prime, this implies that

2

has order

p

in the

multiplicative group

∖

⁡

ℤ

q

{

0

}

and, by Lagrange’s

Theorem

, it divides the order of this group (

http://planetmath.org/Group

), which is

q

-

1

.
∎

Theorem

.

If

m

and

n

are relatively prime

positive integers

, then

2

m

-

1

and

2

n

-

1

are also relatively prime.

Proof.

Let

d

:=

gcd

⁡

(

2

n

-

1

,

2

m

-

1

)

. Since

d

is odd,

2

is a unit in

ℤ

d

and, since

2

n

≡

1

(

mod

d

)

and

2

m

≡

1

(

mod

d

)

, the order of

2

divides both

m

and

n

: it is

1

. Thus

2

≡

1

(

mod

d

)

and

d

=

1

.
∎

Note that these two facts can be easily converted into proofs of the

infinity

of primes: indeed, the first one constructs a prime bigger than any prime

p

and the second easily implies that, if there were finitely many primes, every

M

p

(since there would be as many Mersenne numbers as primes) is a prime power, which is clearly false (consider

M

11

=

23

⋅

89

).

Title

Mersenne numbers, two small results on

Canonical name

MersenneNumbersTwoSmallResultsOn

Date of creation

2013-03-22 15:07:53

Last modified on

2013-03-22 15:07:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Result

Classification

msc 11A41

Related topic

MersenneNumbers

Generated on Fri Feb 9 15:16:50 2018 by

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Mills’ constant

Mills’ constant

Find the smallest real positive number

M

such that

⌊

M

3

n

⌋

is a

prime number

for any

integer

n

>

0

. This is

Mills’

constant

. Assuming that the

Riemann hypothesis

is true, the constant’s value would be approximately 1.3063778838630806904686144926026 (see A051021 in Sloane’s OEIS). According to Caldwell and Cheng, Mills’ original paper “

contained

no numerics,” it only proved the

existence

of such a number; moreover it referred to

⌊

M

c

n

⌋

, and it is those who hunt for the specific value of

M

who often choose

c

=

3

. Armed with these assignments and

assumptions

(including certain

probable primes

), Caldwell and Cheng computed almost seven thousand base 10 digits of Mills’ constant. The first few primes generated by Mills’ constant would be 2, 11, 1361, 2521008887, etc. (listed in A051254).

References

1

C. K. Caldwell & Y. Cheng, “Determining Mills’ Constant and a Note on
Honaker’s Problem”

J.

Integer Sequences

8

(2005): 05.4.1

2

W. H. Mills, “A prime-representing

function

”,

Bull. Amer. Math. Soc.

53

(1947): 604

Title

Mills’ constant

Canonical name

MillsConstant

Date of creation

2013-03-22 16:42:18

Last modified on

2013-03-22 16:42:18

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Synonym

Mills constant

Synonym

Mills’s constant

Generated on Fri Feb 9 15:16:52 2018 by

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minimal prime

minimal prime

A

minimal prime

is a

prime number

p

that when written in a given base

b

, no smaller prime

q

<

p

can be formed from a substring of the digits of

p

(the digits need not be consecutive, but they must be in the same order). For example, in base 10, the prime 991 is a minimal prime because all of its possible substrings (9, 9, 1, 99, 91, 91) are either

composite

or not considered prime. A071062 of Sloane’s OEIS lists the twenty-six base 10 minimal primes.

Clearly, all primes

p

<

b

are minimal primes in that base. Such primes are obviously finite, but so are those minimal primes

p

>

b

, per Michel Lothaire’s findings. In binary, there are only exactly two minimal primes: 2 and 3, written 10 and 11 respectively. Every larger prime will have 1 as its most

significant digit

and possibly a 0 somewhere; the 1 and 0 can then be brought together to form 10 (2 in decimal). The exception to this are the

Mersenne primes

2

q

-

1

(or binary

repunits

), but it is even more elegant to prove these are not minimal primes in binary: they contain all smaller Mersenne primes as substrings!

References

1

M. Lothaire “Combinatorics on words” in

Encylopedia of mathematics and its applications

17

New York: Addison-Wesley (1983): 238 - 247

Title

minimal prime

Canonical

name

MinimalPrime

Date of creation

2013-03-22 16:52:23

Last modified on

2013-03-22 16:52:23

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Classification

msc 11A63

Generated on Fri Feb 9 15:16:54 2018 by

LaTeXML
near-square prime

near-square prime

A

near-square prime

is a

prime number

p

of the form

n

2

+

k

, with

n

being any

integer

and

0

<

|

k

|

<

|

n

|

also an integer. Since for any nonzero real number

x

it is always the case that

x

2

≥

0

, it doesn’t matter if

n

is

negative

.

5

149

4

29

53

3

67

103

2

11

83

1

5

17

37

101

0

1

4

9

16

25

36

49

64

81

100

121

144

-

1

3

-

2

7

23

47

79

-

3

97

-

4

-

5

31

59

139

Fermat primes

are near-square primes for

k

=

1

with the additional requirement that

n

=

2

2

m

-

1

, while Carol primes are near-square primes for

k

=

-

2

with the additional requirement that

n

=

2

m

-

1

.

For

k

=

-

1

, only

n

=

2

gives a prime, namely 3.

Title

near-square prime

Canonical

name

NearsquarePrime

Date of creation

2013-03-22 18:57:37

Last modified on

2013-03-22 18:57:37

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:16:55 2018 by

LaTeXML
partial fractions

partial fractions

Every

fractional number

, i. e. such a

rational number

m

n

that the

integer

m

is not

divisible

by the integer

n

, can be decomposed to a sum of

partial fractions

as follows:

m

n

=

m

1

p

1

ν

1

+

m

2

p

2

ν

2

+

⋯

+

m

t

p

t

ν

t

Here, the

p

i

’s are distinct

positive

prime numbers

, the

ν

i

’s

positive integers

and the

m

i

’s some integers.  Cf. the partial fractions of expressions.

Examples:

6

289

=

6

17

2

-

1

24

=

-

3

2

3

+

1

3

1

1

504

=

-

1

2

3

+

32

3

2

-

24

7

1

How to get the

numerators

m

i

for decomposing a fractional number

1

n

to partial fractions?  First one can take the highest power

p

ν

of a prime

p

which divides the denominator

n

.  Then

n

=

p

ν

⁢

u

,  where

gcd

⁡

(

u

,

p

ν

)

=

1

.  Euclid’s algorithm gives some integers

x

and

y

such that

1

=

x

⁢

u

+

y

⁢

p

ν

.

Dividing this equation by

p

ν

⁢

u

gives the

1

n

=

1

p

ν

⁢

u

=

x

p

ν

+

y

u

.

If

u

has more than one

distinct prime factors

, a

similar

procedure can be made for the fraction

y

u

, and so on.

Note.

The numerators

m

1

,

m

2

, …,

m

t

in the decomposition are not unique.  E. g., we have also

-

1

24

=

-

11

2

3

+

4

3

1

.

Cf. the programme “Murto” (in Finnish) or “Murd” (in Estonian) or “Bruch” (in German) or “Bråk” (in Swedish) or “Fraction”(in French)

http://www.wakkanet.fi/ pahio/ohjelmi.html

here.

Title

partial fractions

Canonical

name

PartialFractions

Date of creation

2013-03-22 14:18:10

Last modified on

2013-03-22 14:18:10

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

34

Author

pahio (2872)

Entry type

Definition

Classification

msc 11A41

Synonym

partial fractions of fractional numbers

Related topic

CategoryOfAdditiveFractions

Defines

fractional number

Generated on Fri Feb 9 15:16:57 2018 by

LaTeXML