aarjaneiro/planetmath huggingface dataset:
palindrome

palindrome

A

palindrome

is a number which yields itself when its digits are reversed. Some palindromes (in base 10) are :

•

121

•

2002

•

314159951413

Clearly one can construct palindromes of arbitrary length by taking any number and appending to it a reversed copy of itself or of all but the last digit.

The concept of palindromes can also be extended to

sequences

and strings.

Title

palindrome

Canonical name

Palindrome

Date of creation

2013-03-22 12:57:05

Last modified on

2013-03-22 12:57:05

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

5

Author

yark (2760)

Entry type

Definition

Classification

msc 11B83

Related topic

Reversal

Generated on Fri Feb 9 15:30:01 2018 by

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proof of Beatty’s theorem

proof of Beatty’s theorem

We define

a

n

:=

n

⁢

p

and

b

n

:=

n

⁢

q

. Since

p

and

q

are irrational, so are

a

n

and

b

n

.

It is also the case that

a

n

≠

b

m

for all

m

and

n

, for if

n

⁢

p

=

m

⁢

q

then

q

=

1

+

n

m

would be

rational

.

The

theorem

is

equivalent

with the statement that for each integer

N

≥

1

exactly

1

element of

{

a

n

}

∪

{

b

n

}

lies in

(

N

,

N

+

1

)

.

Choose

N

integer. Let

s

⁢

(

N

)

be the number of elements of

{

a

n

}

∪

{

b

n

}

less than

N

.

a

n

<

N

⇔

n

⁢

p

<

N

⇔

n

<

N

p

So there are

⌊

N

p

⌋

elements of

{

a

n

}

less than

N

and likewise

⌊

N

q

⌋

elements of

{

b

n

}

.

By definition,

N

p

-

1

<

⌊

N

p

⌋

<

N

p

N

q

-

1

<

⌊

N

q

⌋

<

N

q

and summing these

inequalities

gives

N

-

2

<

s

⁢

(

N

)

<

N

which gives that

s

⁢

(

N

)

=

N

-

1

since

s

⁢

(

N

)

is integer.

The number of elements of

{

a

n

}

∪

{

b

n

}

lying in

(

N

,

N

+

1

)

is then

s

⁢

(

N

+

1

)

-

s

⁢

(

N

)

=

1

.

Title

proof of Beatty’s theorem

Canonical name

ProofOfBeattysTheorem

Date of creation

2013-03-22 13:18:58

Last modified on

2013-03-22 13:18:58

Owner

lieven (1075)

Last modified by

lieven (1075)

Numerical id

8

Author

lieven (1075)

Entry type

Proof

Classification

msc 11B83

Generated on Fri Feb 9 15:30:03 2018 by

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proof that powers of 2 are a superincreasing sequence

proof that powers of 2 are a superincreasing sequence

We will prove a more general fact. If

x

≥

2

then the

sequence

given by

a

k

=

x

k

is a

superincreasing sequence

.

Notice that

a

1

=

x

>

1

=

a

0

. Now we proceed by

induction

.
We assume that

a

n

=

x

n

>

∑

j

=

0

n

-

1

a

j

=

1

+

x

+

x

2

+

x

3

+

⋯

+

x

n

-

1

We need to prove that

a

n

+

1

=

x

n

+

1

>

∑

j

=

0

n

a

j

. But

x

n

+

1

=

x

⋅

x

n

≥

2

⁢

x

n

=

x

n

+

x

n

>

x

n

+

(

x

n

-

1

+

x

n

-

2

⁢

⋯

+

x

+

1

)

so we conclude that

a

n

+

1

>

a

n

+

a

n

-

1

+

⋯

+

a

1

+

a

0

and so the sequence is superincreasing.

Title

proof that powers of 2 are a superincreasing sequence

Canonical name

ProofThatPowersOf2AreASuperincreasingSequence

Date of creation

2013-03-22 15:03:17

Last modified on

2013-03-22 15:03:17

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Proof

Classification

msc 11B83

Generated on Fri Feb 9 15:30:05 2018 by

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Sierpiński conjecture

Sierpiński conjecture

In 1960 Wacław Sierpiński (1882-1969) proved the following interesting result:

Theorem

:

There exist infinitely many

odd integers

k

such that

k

⁢

2

n

+

1

is

composite

for every

integer

n

≥

1

.

A multiplier

k

with this property
is called a Sierpiński number (

http://planetmath.org/SierpinskiNumbers

).
The Sierpiński problem consists in
determining the smallest Sierpiński number.
In 1962, John Selfridge discovered the Sierpiński number

k

=

78557

,
which is now believed to be in fact the smallest such number.

Conjecture:

The integer

k

=

78557

is the smallest Sierpiński number.

To prove the conjecture, it would be

sufficient

to exhibit
a prime

k

⁢

2

n

+

1

for each

k

<

78557

.

Title

Sierpiński conjecture

Canonical name

SierpinskiConjecture

Date of creation

2013-03-22 13:34:16

Last modified on

2013-03-22 13:34:16

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Conjecture

Classification

msc 11B83

Synonym

Sierpinski conjecture

Generated on Fri Feb 9 15:30:07 2018 by

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Sierpinski number

Sierpinski number

An

integer

k

is a

Sierpinski number

if for every

positive integer

n

, the number

k

⁢

2

n

+

1

is

composite

.

That such numbers exist is amazing, and even more surprising is that there are infinitely many of them (in fact, infinitely many odd ones). The smallest

known

Sierpinski number is 78557, but it is not known whether or not this is the smallest one. The smallest number

m

for which it
is unknown whether or not

m

is a Sierpinski number is 10223.

A process for generating Sierpinski numbers using covering sets of primes can be found at

http://www.glasgowg43.freeserve.co.uk/siercvr.htm

Visit

http://www.seventeenorbust.com/

for the

distributed computing

effort to show that 78557 is indeed the smallest Sierpinski number (or find a smaller one).

Similarly, a

Riesel number

is a number

k

such that for every positive integer

n

, the number

k

⁢

2

n

-

1

is composite. The smallest known Riesel number is 509203, but again, it is not known for sure that this is the smallest.

Title

Sierpinski number

Canonical name

SierpinskiNumber

Date of creation

2013-03-22 13:55:33

Last modified on

2013-03-22 13:55:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 11B83

Defines

Riesel number

Defines

Sierpiński number

Generated on Fri Feb 9 15:30:09 2018 by

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square-free sequence

square-free sequence

A

square-free sequence

is a

sequence

which has no

adjacent

repeating

subsequences

of any length.

The name “

square-free

” comes from notation: Let

{

s

}

be a sequence. Then

{

s

,

s

}

is also a sequence, which we write “compactly” as

{

s

2

}

. In the rest of this entry we use a compact notation, lacking commas or braces. This notation is commonly used when dealing with sequences in the capacity of strings. Hence we can write

{

s

,

s

}

=

s

⁢

s

=

s

2

.

Some examples:

•

x

⁢

a

⁢

b

⁢

c

⁢

a

⁢

b

⁢

c

⁢

x

=

x

⁢

(

a

⁢

b

⁢

c

)

2

⁢

x

, not a square-free sequence.

•

a

⁢

b

⁢

c

⁢

d

⁢

a

⁢

b

⁢

c

cannot have any subsequence written in square notation, hence it is a square-free sequence.

•

a

⁢

b

⁢

a

⁢

b

⁢

a

⁢

b

=

(

a

⁢

b

)

3

=

a

⁢

b

⁢

(

a

⁢

b

)

2

, not a square-free sequence.

Note that, while notationally similar to the number-theoretic sense of “square-free,” the two concepts are distinct. For example, for

integers

a

and

b

the product

a

⁢

b

⁢

a

=

a

2

⁢

b

, a square. But as a sequence,

a

⁢

b

⁢

a

=

{

a

,

b

,

a

}

; clearly lacking any commutativity that might allow us to shift elements. Hence, the sequence

a

⁢

b

⁢

a

is square-free.

Title

square-free sequence

Canonical

name

SquarefreeSequence

Date of creation

2013-03-22 11:55:36

Last modified on

2013-03-22 11:55:36

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

12

Author

akrowne (2)

Entry type

Definition

Classification

msc 11B83

Synonym

square free sequence

Generated on Fri Feb 9 15:30:12 2018 by

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stream interlace and deinterlace

stream interlace and deinterlace

Interlace

is the method to create a new

data stream

from
two data streams, each of which has a

constant

time interval

sequence

.
Formally, suppose

A

=

(

a

,

Δ

a

)

and

B

=

(

b

,

Δ

b

)

are two data streams, each have a constant time sequence. For convenience,
we use

Δ

a

and

Δ

b

to also denote the constant terms
of each of those sequences. We construct a new data stream

C

=

(

c

,

Δ

c

)

, also having constant time interval, as follows:

c

n

=

{

b

n

-

⌊

n

⁢

z

⌋

⌊

n

⁢

z

⌋

=

⌊

(

n

+

1

)

⁢

z

⌋

a

⌊

n

⁢

z

⌋

⌊

n

⁢

z

⌋

≠

⌊

(

n

+

1

)

⁢

z

⌋

,

z

=

Δ

b

Δ

a

+

Δ

b

,

Δ

c

=

Δ

a

⁢

Δ

b

Δ

a

+

Δ

b

Deinterlace

is the method of constructing two data streams ,

A

and

B

,
each having constant time interval, from a given data stream

C

and

primary

interlace value

Δ

of computed stream, where

C

has constant time interval.

a

n

=

c

n

+

⌈

(

n

+

1

)

⁢

Δ

a

Δ

b

⌉

,

Δ

a

=

Δ

c

⁢

Δ

b

|

Δ

c

-

Δ

b

|

and

b

n

=

c

n

+

⌊

n

⁢

Δ

b

Δ

a

⌋

,

Δ

b

=

Δ

c

⁢

Δ

a

|

Δ

c

-

Δ

a

|

This sequences are the

Fraenkel partition theorem

instance

.

References

[1

] Aviezri S. Fraenkel,

The

bracket function

and complementary sets of integers

, Canad. J.
Math.

21

(1969), 6–27.

http://www.ams.org/mathscinet-getitem?mr=38:3214

MR
38:3214

[2

] Michal Widera,

Deterministic

method of data sequence processing

, Vol. IV, ISSN 1732-1360, Annales UMCS (2006), 314–331.

http://www.annales.umcs.lublin.pl/AI/index.html

UMCS Annales AI

Title

stream interlace and deinterlace

Canonical name

StreamInterlaceAndDeinterlace

Date of creation

2013-03-22 15:37:08

Last modified on

2013-03-22 15:37:08

Owner

michal (7107)

Last modified by

michal (7107)

Numerical id

16

Author

michal (7107)

Entry type

Theorem

Classification

msc 11B83

Synonym

interlace deinterlace

Related topic

FraenkelsPartitionTheorem

Related topic

BeattySequence

Related topic

DataStream

Defines

stream junction method

Generated on Fri Feb 9 15:30:15 2018 by

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superincreasing sequence

superincreasing sequence

A

sequence

{

s

j

}

of real numbers is

superincreasing

if

s

n

+

1

>

∑

j

=

1

n

s

j

for every

positive integer

n

. That is, any element of the sequence is greater than all of the previous

elements

added together.

A commonly used superincreasing sequence is that of

powers of two

(

s

n

=

2

n

.)

Suppose that

x

=

∑

j

=

1

n

a

j

⁢

s

j

. If

{

s

j

}

is a superincreasing sequence and every

a

j

∈

{

0

,

1

}

, then we can always determine the

a

j

’s simply by knowing

x

. This is analogous to the fact that, for any

natural number

, we can always determine which bits are on and off in the binary bitstring representing the number.

Title

superincreasing sequence

Canonical

name

SuperincreasingSequence

Date of creation

2013-03-22 11:55:22

Last modified on

2013-03-22 11:55:22

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11B83

Synonym

superincreasing

Related topic

Superconvergence

Generated on Fri Feb 9 15:30:17 2018 by

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Prouhet-Thue-Morse sequence

Prouhet-Thue-Morse sequence

The

Prouhet-Thue-Morse sequence

is a binary

sequence

which begins as follows:

0

,

1

,

1

,

0

,

1

,

0

,

0

,

1

,

1

,

0

,

0

,

1

,

0

,

1

,

1

,

0

,

…

The

n

th term is defined to be the number of

1

s in the binary expansion of

n

, modulo 2. That is,

t

n

=

0

if the number of

1

s in the binary expansion of

n

is even, and

t

n

=

1

if it is odd.

The sequence

satisfies

the following

recurrence relation

, with

t

0

=

0

:

t

2

⁢

n

=

t

n

t

2

⁢

n

+

1

=

1

-

t

n

The Prouhet-Thue-Morse sequence is an automatic sequence. It has been shown to be

(no three consecutive identical blocks) and overlap-free
i.e no sub-block of the form

a

⁢

w

⁢

a

⁢

w

⁢

a

, where

a

∈

{

0

,

1

}

, when viewed as a word of

infinite

length over the binary alphabet

{

0

,

1

}

.

Generating function

The

generating function

T

⁢

(

x

)

=

∑

n

=

0

∞

t

n

⁢

x

n

for the sequence satisfies the

relation

T

⁢

(

x

)

=

T

⁢

(

x

2

)

⁢

(

1

-

x

)

+

x

1

-

x

2

History

The Thue-Morse sequence was independently discovered by P. Prouhet, Axel Thue, and Marston Morse, and has since been rediscovered by many others.

References

•

Allouche, J.-P.; Shallit, J. O.

http://www.cs.uwaterloo.ca/ shallit/Papers/ubiq.ps

The ubiquitous Prouhet-Thue-Morse Sequence [postscript]

•

Sloane, N. J. A.

Sequence A010060

,

http://www.research.att.com/ njas/sequences/

The

On-Line Encyclopedia of Integer Sequences

.

Title

Prouhet-Thue-Morse sequence

Canonical

name

ProuhetThueMorseSequence

Date of creation

2013-03-22 14:27:17

Last modified on

2013-03-22 14:27:17

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

24

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 11B85

Classification

msc 68R15

Synonym

Thue-Morse sequence

Related topic

ProuhetThueMorseConstant

Generated on Fri Feb 9 15:30:20 2018 by

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closed form

closed form

A

closed form

function

which gives the value of a

sequence

at index

n

has only one parameter,

n

itself. This is in contrast to the

recurrence relation

form, which can have all of the previous values of the sequence as parameters.

The benefit of the closed form is that one does not have to

calculate

all of the previous values of the sequence to get the next value. This is not too useful if one wants to print out or utilize all of the values of a sequence up to some

n

, but it is very useful to get the value of the sequence

just

at some index

n

.

There are many techniques used to find a closed-form solution for a recurrence relation. Some are

•

Repeated

substitution

. Replace each

s

k

in the expression of

s

n

(with

k

<

n

) with its recurrence relation representation. Repeat again on the resulting expression, until some pattern is evident.

•

Estimate an

upper bound

for

s

n

in terms of

n

. Then, solve for the unknowns (say there are

r

unknowns) by finding the first

r

values of the recurrence relation and solving the

linear system

formed by them and the unknowns.

•

Find the

characteristic equation

of the recurrence relation and solve for the roots. If the recurrence relation is not

homogeneous

, then you’ll have to apply a method such as the

method of undetermined coefficients

.

Title

closed form

Canonical name

ClosedForm

Date of creation

2013-03-22 11:56:10

Last modified on

2013-03-22 11:56:10

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 11B99

Synonym

closed-form

Related topic

ExpressibleInClosedForm

Generated on Fri Feb 9 15:30:27 2018 by

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example of closed form

example of closed form

Consider the recurrence given by

x

0

=

7

and

x

n

=

2

⁢

x

n

-

1

. Then it can be proved by

induction

that

x

n

=

2

n

⋅

7

.

The

expression

x

n

=

2

n

⋅

7

is a

closed form

expression the recurrence given, since it depends exclusively on

n

, whereas the recurrence depends on

n

and

x

n

-

1

(the previous value).

Now consider Fibonacci’s recurrence:

f

n

=

f

n

-

1

+

f

n

-

2

,

f

0

=

1

.

It is not a

closed formula

, since if we wanted to compute

f

10

we would need to know

f

9

,

f

8

(and for knowing them, the previous terms too). However, such recurrence has a closed formula, known as

Binet formula

:

f

n

=

(

1

+

5

2

)

n

-

(

1

-

5

2

)

n

5

.

Binet formula is closed, since if we wanted to compute

f

n

we only need to substitute

n

on the previous

formula

, and no additional information is required.

Title

example of closed form

Canonical

name

ExampleOfClosedForm

Date of creation

2013-03-22 15:03:07

Last modified on

2013-03-22 15:03:07

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Example

Classification

msc 11B99

Generated on Fri Feb 9 15:30:29 2018 by

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Lychrel number

Lychrel number

A

Lychrel number

is a number which never yields a

palindrome

in the iterative process of adding to itself a copy of itself with digits reversed. For example, if we start with the number 983 we get:

•

983

+

389

=

1372

•

1372

+

2731

=

4103

•

4103

+

3014

=

7117

So in 3 steps we get a palindrome, hence 983 is not a Lychrel number.

In fact, it is not known if there exist any Lychrel numbers in base 10 (numbers colloquially called “Lychrel numbers” in base 10 are in fact just

Lychrel candidates

). However, in base 2 for example, there have been numbers proven to be Lychrel numbers

1

1

[

2

]

informs us that Ronald Sprague has proved that the number 10110, for example, is a Lychrel number is base 2.

. The first Lychrel candidate is 196:

•

196

+

691

=

887

•

887

+

788

=

1675

•

1675

+

5761

=

7436

•

7436

+

6347

=

13783

•

13783

+

38731

=

52514

•

52514

+

41525

=

94039

•

94039

+

93049

=

187088

•

187088

+

880781

=

1067869

•

…

This has been followed out to millions of digits, with no palindrome found in the

sequence

.

The following table gives the number of Lychrel candidates found within ascending ranges:

Range

Possible Lychrels

0 - 100

0

100 - 1,000

2

1,000 - 10,000

3

10,000 - 100,000

69

100,000 - 1,000,000

99

10,000,000 - 100,000,000

1728

100,000,000 - 1,000,000,000

29,813

References

1

Wade VanLandingham,

http://www.p196.org/

196 And Other Lychrel Numbers

2

John Walker,

http://www.fourmilab.ch/documents/threeyears/threeyears.html

Three Years of Computing

Title

Lychrel number

Canonical name

LychrelNumber

Date of creation

2013-03-22 12:57:09

Last modified on

2013-03-22 12:57:09

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

10

Author

akrowne (2)

Entry type

Definition

Classification

msc 11B99

Generated on Fri Feb 9 15:30:31 2018 by

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A. Cohn’s irreducibility criterion

A. Cohn’s irreducibility criterion

Theorem

.

Assume

n

≥

2

is an integer and that

P

is a

polynomial

with

coefficients

in

{

0

,

1

,

…

,

n

-

1

}

. If

P

⁢

(

n

)

is prime then

P

⁢

(

x

)

is

irreducible

(

http://planetmath.org/IrreduciblePolynomial2

) in

Z

⁢

[

x

]

.

A proof is given in

[

MRM

]

.

A. Cohn

[

PZ

]

proved this theorem for the case

n

=

10

.

This special case of the above theorem is sketched as problem 128, Part VIII, in

[

PZ

]

.

References

PZ

George PÃÂ³lya, Gabor Szego,

Problems and Theorems in Analysis II

,
Classics in Mathematics 1998.

MRM

M. Ram Murty,

Prime Numbers and

Irreducible Polynomials

, American
Mathematical Monthly, vol. 109, (2002), 452-458.

Title

A. Cohn’s irreducibility criterion

Canonical name

ACohnsIrreducibilityCriterion

Date of creation

2013-03-22 14:37:02

Last modified on

2013-03-22 14:37:02

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

17

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 11C08

Generated on Fri Feb 9 15:30:33 2018 by

LaTeXML
A property of polynomials

A property of polynomials

A

⁢

P

⁢

R

⁢

O

⁢

P

⁢

E

⁢

R

⁢

T

⁢

Y

⁢

O

⁢

F

⁢

P

⁢

O

⁢

L

⁢

Y

⁢

N

⁢

O

⁢

M

⁢

I

⁢

A

⁢

L

⁢

S

Let

f

⁢

(

x

)

be a

polynomial

in

x

where

x

a

⁢

n

⁢

d

⁢

t

⁢

h

⁢

e

⁢

c

⁢

o

⁢

e

⁢

f

⁢

f

⁢

i

⁢

c

⁢

i

⁢

e

⁢

n

⁢

t

⁢

s

a

)

a

r

e

i

n

t

e

g

e

r

s

,

b)

x

i

⁢

s

⁢

a

⁢

s

⁢

q

⁢

u

⁢

a

⁢

r

⁢

e

⁢

m

⁢

a

⁢

t

⁢

r

⁢

i

⁢

x

⁢

a

⁢

n

⁢

d

k

.5

⁢

i

⁢

n

⁢

i

⁢

s

⁢

a

⁢

n

⁢

a

⁢

t

⁢

u

⁢

r

⁢

a

⁢

l

⁢

n

⁢

u

⁢

m

⁢

b

⁢

e

⁢

r

.

T

⁢

h

⁢

e

⁢

p

⁢

r

⁢

o

⁢

p

⁢

e

⁢

r

⁢

t

⁢

y

:

f

⁢

(

x

+

k

*

f

⁢

(

x

)

)

⁢

i

⁢

s

⁢

c

⁢

o

⁢

n

⁢

g

⁢

r

⁢

u

⁢

e

⁢

n

⁢

t

⁢

t

⁢

o

0

m

⁢

o

⁢

d

f(x)

.

Proof:

$

T

h

e

r

e

i

s

n

o

l

o

s

s

o

f

g

e

n

e

r

a

l

i

t

y

i

n

t

a

k

i

n

g

k

=

1

.

B

y

T

a

y

l

o

r

′

s

t

h

e

o

r

e

m

w

e

g

e

t

f

(

x

+

f

(

x

)

)

=

f

(

x

)

+

f

(

x

)

f

′

(

x

)

+

(

f

(

x

)

2

)

/

2

f

′′

(

x

)

+

(

f

(

x

)

3

)

/

3

!

f

′′′

(

x

)

…

.

=

f

(

x

)

1

+

f

′

(

x

)

+

f

(

x

)

/

2

*

f

′′

(

x

)

+

f

(

x

)

2

/

3

!

.

.

i

.

e

.

f

(

x

+

f

(

x

)

)

i

s

c

o

n

g

r

u

e

n

t

t

o

0

(

m

o

d

(

f

(

x

)

)

)

.

Title

A property of polynomials

Canonical name

APropertyOfPolynomials

Date of creation

2013-03-22 19:34:45

Last modified on

2013-03-22 19:34:45

Owner

akdevaraj (13230)

Last modified by

akdevaraj (13230)

Numerical id

17

Author

akdevaraj (13230)

Entry type

Definition

Classification

msc 11C08

Generated on Fri Feb 9 15:30:35 2018 by

LaTeXML
binomial equation

binomial equation

Binomial equation

is an

algebraic equation

of the

x

n

-

a

=

0

where

n

is a

positive integer

,

a

belongs to a certain field (or sometimes to a certain ring) and

x

is the unknown (or the indeterminate) of the equation.  Solving the binomial equation means taking the

n

th root (

http://planetmath.org/NthRoot

) of

a

.

The binomial equation is written also

x

n

=

a

.

Such a binomial equation may be examined in a group, too.

A special case of the binomial equation is the

cyclotomic equation

x

n

-

1

=

0

.

This name comes from the fact that the roots (

http://planetmath.org/Equation

) of the equation the

unit circle

in the

complex plane

into

n

equally long arcs (Greek

ϰ

⁢

υ

´

⁢

ϰ

⁢

λ

⁢

o

⁢

ς

‘circle’,

τ

⁢

o

´

⁢

μ

⁢

o

⁢

ς

‘part’).

Title

binomial equation

Canonical name

BinomialEquation

Date of creation

2013-03-22 17:46:05

Last modified on

2013-03-22 17:46:05

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 11C08

Classification

msc 12E05

Related topic

Binomial

Related topic

CalculatingTheNthRootsOfAComplexNumber

Related topic

RootOfUnity

Defines

cyclotomic equation

Generated on Fri Feb 9 15:30:37 2018 by

LaTeXML
content of a polynomial

content of a polynomial

Let

P

=

a

0

+

a

1

⁢

x

+

…

+

a

n

⁢

x

n

∈

ℤ

⁢

[

x

]

be a

polynomial

with

integer

coefficients. The content of

P

is the

greatest common divisor

of the coefficients of

P

.

c

⁢

(

P

)

=

gcd

⁡

(

a

0

,

a

1

,

…

,

a

n

)

Title

content of a polynomial

Canonical name

ContentOfAPolynomial

Date of creation

2013-03-22 13:33:42

Last modified on

2013-03-22 13:33:42

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

4

Author

Daume (40)

Entry type

Definition

Classification

msc 11C08

Related topic

ProductOfFinitelyGeneratedIdeals

Generated on Fri Feb 9 15:30:38 2018 by

LaTeXML