aarjaneiro/planetmath huggingface dataset:
Erdős-Fuchs theorem

Erdős-Fuchs theorem

Let

A

be a set of natural numbers. Let

R

n

⁢

(

A

)

be the number of ways to represent

n

as a sum of two elements in

A

, that is,

R

n

⁢

(

A

)

=

∑

a

i

+

a

j

=

n

a

i

,

a

j

∈

A

1

.

Erdős-Fuchs

theorem

[

1

,

2

]

states that if

c

>

0

, then

∑

n

≤

N

R

n

⁢

(

A

)

=

c

⁢

N

+

o

⁢

(

N

1

4

⁢

log

-

1

2

⁡

N

)

(1)

cannot hold.

On the other hand, Ruzsa

[

5

]

constructed a set for which

∑

n

≤

N

R

n

⁢

(

A

)

=

c

⁢

N

+

O

⁢

(

N

1

4

⁢

log

⁡

N

)

.

Montgomery and Vaughan

[

4

]

improved on the original Erdős-Fuchs theorem by showing that for every

c

>

0

max

N

≤

M

⁡

|

∑

n

≤

N

R

n

⁢

(

A

)

-

c

⁢

n

|

=

Ω

⁢

(

M

1

4

⁢

log

-

1

4

⁡

M

)

holds. In

[

4

]

a result of Jurkat is cited
which appeared
in the Ph. D. thesis of Hayashi

[

3

]

which
improves

N

1

4

⁢

log

-

1

2

⁡

N

in (

1

) to

N

1

4

.

References

1

Paul Erdős and Wolfgang H.J. Fuchs.

On a problem of

additive number theory

.

J. Lond. Math. Soc.

, 31:67–73, 1956.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0070.04104

Zbl 0070.04104.

2

Heini Halberstam and Klaus Friedrich Roth.

Sequences

.

Springer-Verlag, second edition, 1983.

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

Zbl 0498.10001.

3

E. K. Hayashi.

Omega theorems for the iterated

additive

convolution of
non-negative

arithmetic function

.

PhD thesis, University of Illinois at Urbana-Champaign, 1973.

4

H. L. Montgomery and R. C. Vaughan.

On the Erdős-Fuchs theorems.

In

A tribute to Paul Erdős

, pages 331–338. Cambridge Univ.
Press, Cambridge, 1990.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0715.11005

Zbl 0715.11005.

5

Imre Ruzsa.

A converse to a theorem of Erdős and Fuchs.

J.

Number Theory

, 62(2):397–402, 1997.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0872.11014

Zbl 0872.11014.

Title

Erdős-Fuchs theorem

Canonical

name

ErdHosFuchsTheorem

Date of creation

2013-03-22 13:20:59

Last modified on

2013-03-22 13:20:59

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

12

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11B34

Generated on Fri Feb 9 15:26:49 2018 by

LaTeXML
Collatz problem

Collatz problem

We define the function

f

:

ℕ

⟶

ℕ

(where

ℕ

excludes zero) such that

f

⁢

(

a

)

=

{

3

⁢

a

+

1

if

⁢

a

⁢

is odd

a

/

2

if

⁢

a

⁢

is even.

Then let the

sequence

c

n

be defined as

c

i

=

f

⁢

(

c

i

-

1

)

, with

c

0

an arbitrary natural seed value.

It is conjectured that the sequence

c

0

,

c

1

,

c

2

,

…

will always end in

1

,

4

,

2

, repeating infinitely. This has been verified by computer up to very large values of

c

0

, but is unproven in general. It is also not known whether this problem is decideable. This is generally called the

Collatz problem

.

The sequence

c

n

is sometimes called the “hailstone sequence”. This is because it behaves analogously to a hailstone in a cloud which falls by gravity and is tossed up again repeatedly. The sequence similarly ends in an eternal oscillation.

Title

Collatz problem

Canonical name

CollatzProblem

Date of creation

2013-03-22 11:42:43

Last modified on

2013-03-22 11:42:43

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

32

Author

akrowne (2)

Entry type

Conjecture

Classification

msc 11B37

Synonym

Ulam’s Problem

Synonym

1-4-2 Problem

Synonym

Syracuse problem

Synonym

Thwaites conjecture

Synonym

Kakutani’s problem

Synonym

3n+1 problem

Generated on Fri Feb 9 15:26:57 2018 by

LaTeXML
Collatz sequence

Collatz sequence

A

Collatz sequence

is a

sequence

formed by iteratively applying the function defined for the

Collatz problem

to a given starting

integer

n

, in which if

2

|

n

,

f

⁢

(

n

)

=

n

2

and if not then

f

⁢

(

n

)

=

3

⁢

n

+

1

.

For example, the Collatz sequence starting with 47 goes: 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1.

It is

obvious

that for a power of 2,

n

=

2

x

, the Collatz sequence will be

x

long (not counting the starting number) and consist of the first

x

integer powers of 2 in

descending order

:

2

x

-

1

,

2

x

-

2

,

…

,

2

,

1

.

For other kinds of

n

there are various formulas giving heuristic estimates of the length of the Collatz sequence of

n

but no simple formula to give the exact value. The number of iterations needed to reach 1 for the first few

n

are 1, 2, 8, 3, 6, 9, 17, 4, 20, 7, 15, 10, 10, 18, 18, 5, 13, 21, 21, 8, 8, 16, 16, 11, 24, etc., listed in sequence A008908 of Sloane’s OEIS. The number of iterations needed to reach a power of 2 for the first few

n

are 0, 0, 3, 0, 1, 4, 12, 0, 15, 2, 10, 5, 5, 13, 13, 0, 8, 16, 16, 3, 1, 11, 11, 6, 19, 6, 107, 14, 14, 14, 102, 0, 22, 9, 9, 17, 17, 17, 30, 4, 105, 2, 25, etc., which can be calculated with the Mathematica command

Flatten[Table[Take[Select[Range[Length[Collatz[n]]], IntegerQ[1 / Log[Collatz[n][[#]], 2]] &], 1], {n, 100}]]

(the

Collatz[n]

command needs to be defined by the user; the program will complain about

division by zero

but still give the desired results).

Title

Collatz sequence

Canonical

name

CollatzSequence

Date of creation

2013-03-22 16:43:51

Last modified on

2013-03-22 16:43:51

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B37

Generated on Fri Feb 9 15:26:59 2018 by

LaTeXML
cube of an integer

cube of an integer

Theorem

.

Any

cube of integer

is a difference of two squares, which in the case
of a

positive

cube are the squares of two successive

triangular numbers

.

For proving the assertion, one needs only to check the identity

a

3

≡

(

(

a

+

1

)

⁢

a

2

)

2

-

(

(

a

-

1

)

⁢

a

2

)

2

.

For example we have

(

-

2

)

3

=

1

2

-

3

2

and

4

3

=

64

=

10

2

-

6

2

.

Summing

the first

n

positive cubes, the identity allows

http://planetmath.org/encyclopedia/TelescopingSum.html

telescoping between consecutive brackets,

1

3

+

2

3

+

3

3

+

4

3

+

…

+

n

3

=

[

1

2

-

0

2

]

+

[

3

2

-

1

2

]

+

[

6

2

-

3

2

]

+

[

10

2

-

6

2

]

+

…

+

[

(

(

n

+

1

)

⁢

n

2

)

2

-

(

(

n

-

1

)

⁢

n

2

)

2

]

=

(

n

2

+

n

2

)

2

,

saving only the square

(

(

n

+

1

)

⁢

n

2

)

2

.  Thus we have this

expression

presenting the sum of the first

n

positive cubes (cf. the

Nicomachus theorem

).

Title

cube of an

integer

Canonical

name

CubeOfAnInteger

Date of creation

2013-03-22 19:34:33

Last modified on

2013-03-22 19:34:33

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11B37

Classification

msc 11A25

Related topic

NicomachusTheorem

Related topic

TriangularNumbers

Related topic

DifferenceOfSquares

Generated on Fri Feb 9 15:27:06 2018 by

LaTeXML
linear recurrence

linear recurrence

A

sequence

x

0

,

x

1

,

…

,

is said to

satisfy

a

general

linear recurrence

if there are

constants

a

n

,

i

and

f

n

such that

x

n

=

∑

i

=

1

n

a

n

,

i

⁢

x

n

-

i

+

f

n

The sequence

{

f

n

}

is called the

non-homogeneous part

of the recurrence.

If

f

n

=

0

for all

n

then the recurrence is said to be

homogeneous

.

In this case if

x

and

x

′

are two

solutions

to the recurrence,
and

A

and

B

are constants, then

A

⁢

x

+

B

⁢

x

′

are also solutions.

If

a

n

,

i

=

a

i

for all

i

then the recurrence is called
a

linear recurrence with constant coefficients

. Such a recurrence
with

a

n

,

i

=

0

for all

i

>

k

and

a

k

≠

0

is said to be of

finite order

and the smallest such

integer

k

is called the

order

of the recurrence. If no such

k

exists, the recurrence is
said to be of

infinite order

.

For example, the

Fibonacci sequence

is a linear recurrence with constant coefficients and order 2.

Now suppose that

x

=

{

x

n

}

satisfies a homogeneous linear recurrence of
finite order

k

. The

polynomial

F

x

=

u

k

-

a

1

⁢

u

k

-

1

-

…

-

a

k

is called the

companion polynomial

of

x

.
If we assume that the coefficients

a

i

come from a field

K

we can
write

F

x

=

(

u

-

u

1

)

e

1

⁢

⋯

⁢

(

u

-

u

t

)

e

t

where

u

1

,

…

,

u

t

are the distinct roots of

F

x

in some

splitting field

of

F

x

containing

K

and

e

1

,

…

,

e

t

are

positive integers

.
A

theorem

about such

recurrence relations

is that

x

n

=

∑

i

=

1

t

g

i

⁢

(

n

)

⁢

u

i

n

for some polynomials

g

i

in

K

⁢

[

X

]

where

d

⁢

e

⁢

g

⁢

(

g

i

)

<

e

i

.

Now suppose further that all

K

=

ℂ

and

let

U

be the largest value of the

|

u

i

|

.

It follows that if

U

>

1

then

|

x

n

|

≤

C

⁢

U

n

for some constant

C

.

(to be continued)

Title

linear recurrence

Canonical name

LinearRecurrence

Date of creation

2013-03-22 16:52:29

Last modified on

2013-03-22 16:52:29

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 11B37

Generated on Fri Feb 9 15:27:16 2018 by

LaTeXML
maple implementation of Berlekamp-Massey algorithm

maple implementation of Berlekamp-Massey algorithm

\PMlinkescapetext{
>
# Maple code for the Berlekamp-Massey algorithm
# Adapted from www.cs.wisc.edu/~cs435-1/bermas.m
# Transliteration of
# Massey, "Shift-Register Synthesis and BCH Decoding,"
# IEEE Trans. Inform. Theory, 15(1):122-127, 1969.
# Input: P, either 0 or a prime
# If P>0 then we work over the field K = Z/Z[P] (mod P)
# else we work over the field K = Q (rationals)
# N, a positive integer
# s, a list of >= 2*N terms in K
# x, a formal variable
# Returns: Unique monic annihilator of minimum degree, over K[x].

BM := proc(s, N, P, x)
local C,B,T,L,k,i,n,d,b,safemod;
ASSERT(nops(s) = 2*N);
safemod := (exp, P) -> `if`(P=0, exp, exp mod P);
B := 1;
C := 1;
L := 0;
k := 1;
b := 1;
for n from 0 to 2*N-1 do
d := s[n+1];
for i from 1 to L do
d := safemod(d + coeff(C,x^i)*s[n-i+1], P);
od;
if d=0 then k := k+1 fi;
if (d <> 0 and 2*L > n) then
C := safemod(expand(C - d*x^k*B/b), P);
k := k+1;
fi;
if (d <> 0 and 2*L <= n) then
T := C;
C := safemod(expand(C - d*x^k*B/b), P);
B := T;
L := n+1-L;
k := 1;
b := d;
fi;
od;
return C;
end:
}

The following test demonstrates usage and verifies that this works:

\PMlinkescapetext{
> P := 103:
d := 4:
num := 21+83*x+90*x^2+4*x^3: # degree < d
den := 1+11*x+23*x^2+58*x^3+69*x^4: # monic, degree <= d
f := series(num/den, x=0, 2*d) mod P:
s := [seq(coeff(f, x, i), i=0..2*d-1)]:
BM(s, d, P, x);}

1

+

11

⁢

x

+

23

⁢

x

2

+

58

⁢

x

3

+

69

⁢

x

4

The annihilator is the same as

denominator

, as we expect.

Title

maple implementation of Berlekamp-Massey algorithm

Canonical name

MapleImplementationOfBerlekampMasseyAlgorithm

Date of creation

2013-03-22 15:35:27

Last modified on

2013-03-22 15:35:27

Owner

daveagp (6096)

Last modified by

daveagp (6096)

Numerical id

6

Author

daveagp (6096)

Entry type

Algorithm

Classification

msc 11B37

Classification

msc 15A03

Related topic

BerlekampMasseyAlgorithm

Generated on Fri Feb 9 15:27:18 2018 by

LaTeXML
p⁢x+1 map

p

⁢

x

+

1

map

The

p

⁢

x

+

1

map

is a

generalization

of the

3

⁢

x

+

1

map, or

Collatz problem

, in which an input number

x

is tested for

divisibility by primes

less than an

odd prime

p

: if

x

is

divisible

by any such prime, it is divided by the smallest such prime; otherwise it is multiplied by

p

and incremented by 1. In the case

p

=

3

, the test for the input number simplifies to a parity test, because 2 is the only prime less than 3. The map can be iteratively applied, generally resulting in

sequences

that rise and fall aperiodically before reaching 1 in ways that are not readily apparent from the original

x

(with the obvious exception of

x

being a

power of two

, in which case the resulting sequence from the original

x

to 1 is a listing of powers of two in

strictly descending order

; a

similar

pattern occurs when

x

is a power of a prime). Once the sequence reaches 1, continued iteration produces a predictable cycle, especially when

p

is a

Mersenne prime

.

For example, with the

7

⁢

x

+

1

map, an input

x

is tested for

divisibility

by 2, 3 and 5 (in that order), and whichever test passes first causes the output to be

x

divided by that prime; but if all tests fail, then the output is

7

⁢

x

+

1

. Applied to 19, the map gives the sequence 19, 134, 67, 470, 235, 47, 330, 165, 55, 11, 78, 39, 13, 92, 46, 23, 162, 81, 27, 9, 3, 1. (Notice that 470 is followed by 235 and then 47; similarly 330 is followed by 165, 55 and then 11). It is believed that for any

p

, iteration on any

x

will

eventually

reach 1, but just as the original

p

=

3

problem remains unsolved, so does the generalized problem.

Title

p

⁢

x

+

1

map

Canonical name

Px1Map

Date of creation

2013-03-22 18:56:32

Last modified on

2013-03-22 18:56:32

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B37

Generated on Fri Feb 9 15:27:20 2018 by

LaTeXML
theorem on Collatz sequences starting with Mersenne numbers

theorem on Collatz sequences starting with Mersenne numbers

Theorem

.

Given a

Mersenne number

m

=

2

n

-

1

(with

n

a

nonnegative integer

), the

Collatz sequence

starting with

m

reaches

3

n

-

1

in precisely

2

⁢

n

steps. Also, the parity of such a

sequence

consistenly alternates parity until

3

n

-

1

is reached. For example, given

2

2

-

1

=

3

gives the Collatz sequence 3, 10, 5, 16, 8, 4, 2, 1, in which

3

2

-

1

=

8

is reached at the fourth step. Also, the least significant bits of this particular sequence are 1, 0, 1, 0, 0, 0, 0, 1.

As you might already know, a Collatz sequence results from the repeated application of the Collatz

function

C

⁢

(

n

)

=

3

⁢

n

+

1

for odd

n

and

C

⁢

(

n

)

=

n

2

for even

n

. If I may, I’d like to introduce the iterated Collatz function notation as a

recurrence relation

thus:

C

0

⁢

(

n

)

=

n

and

C

i

⁢

(

n

)

=

C

⁢

(

C

i

-

1

⁢

(

n

)

)

for all

i

>

0

. In our example,

C

0

⁢

(

3

)

=

3

,

C

1

⁢

(

3

)

=

10

,

C

2

⁢

(

3

)

=

5

, etc. (We could choose to have

C

1

⁢

(

n

)

=

n

instead with only slight changes to the theorem and its proof).

Proof.

Obviously

m

=

C

0

⁢

(

2

n

-

1

)

=

2

n

-

1

is an

odd number

. Therefore

C

1

⁢

(

m

)

=

2

n

⁢

3

-

2

, an

even number

, and then

C

2

⁢

(

m

)

=

2

n

-

1

⁢

3

-

1

,

C

3

⁢

(

m

)

=

2

n

-

1

⁢

9

-

2

,

C

4

⁢

(

m

)

=

2

n

-

2

⁢

9

-

1

,

C

5

⁢

(

m

)

=

2

n

-

2

⁢

27

-

2

, etc. We can now generalize that if

i

is odd, then

C

i

⁢

(

m

)

=

2

n

-

i

-

1

2

⁢

3

i

+

1

2

-

2

and

C

i

⁢

(

m

)

=

2

n

-

i

2

⁢

3

i

2

if

i

is even. By plugging in

i

=

2

⁢

n

, we get

C

i

⁢

(

m

)

=

2

n

-

2

⁢

n

2

⁢

3

2

⁢

n

2

=

2

n

-

n

⁢

3

n

-

1

=

2

0

⁢

3

n

-

1

=

3

n

-

1

, proving the theorem.
∎

Of course the generalized

formulas

do not work when

i

>

2

⁢

n

, nor does any of this give any insight into when a Collatz sequence starting with a Mersenne number reaches a power of 2. Likewise, the pattern of consistently alternating parity usually breaks down on or right after the

2

⁢

n

th step.

Title

theorem on Collatz sequences starting with Mersenne numbers

Canonical name

TheoremOnCollatzSequencesStartingWithMersenneNumbers

Date of creation

2013-03-22 17:34:32

Last modified on

2013-03-22 17:34:32

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

12

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 11B37

Generated on Fri Feb 9 15:27:22 2018 by

LaTeXML
Catalan’s identity

Catalan’s identity

Let

f

n

be the

n

th

Fibonacci number

n

>

1

, then the following identity is called

Catalan’s identity

.

f

n

2

-

f

n

+

r

⁢

f

n

-

r

=

(

-

1

)

n

-

r

⁢

f

r

2

.

For

r

=

1

we have Cassini’s identity.

Title

Catalan’s identity

Canonical name

CatalansIdentity

Date of creation

2013-03-22 14:42:56

Last modified on

2013-03-22 14:42:56

Owner

vmoraru (1243)

Last modified by

vmoraru (1243)

Numerical id

7

Author

vmoraru (1243)

Entry type

Theorem

Classification

msc 11B39

Related topic

ProofofCassinisIdentity

Generated on Fri Feb 9 15:27:24 2018 by

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derivation of Binet formula

derivation of Binet formula

The

characteristic polynomial

for the Fibonacci recurrence

f

n

=

f

n

-

1

+

f

n

-

2

is

x

2

=

x

+

1

.

The solutions of the

characteristic equation

x

2

-

x

-

1

=

0

are

ϕ

=

1

+

5

2

,

ψ

=

1

-

5

2

so the closed formula for the

Fibonacci sequence

must be of the form

f

n

=

u

⁢

ϕ

n

+

v

⁢

ψ

n

for some

real numbers

u

,

v

. Now we use the

boundary conditions

of the recurrence, that is,

f

0

=

0

,

f

1

=

1

, which means we have to solve the system

0

=

u

⁢

ϕ

0

+

v

⁢

ψ

0

,

1

=

u

⁢

ϕ

1

+

v

⁢

ψ

1

The first equation simplifies to

u

=

-

v

and substituting into the second one gives:

1

=

u

⁢

(

1

+

5

2

)

-

u

⁢

(

1

-

5

2

)

=

u

⁢

(

2

⁢

5

2

)

=

u

⁢

5

.

Therefore

u

=

1

5

,

v

=

-

1

5

and so

f

n

=

ϕ

n

5

-

ψ

n

5

=

ϕ

n

-

ψ

n

5

.

Title

derivation of Binet formula

Canonical name

DerivationOfBinetFormula

Date of creation

2013-03-22 15:03:50

Last modified on

2013-03-22 15:03:50

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Derivation

Classification

msc 11B39

Generated on Fri Feb 9 15:27:28 2018 by

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derivation of plastic number

derivation of plastic number

The

plastic number

may be defined to be the limit of the ratio of two successive members (

http://planetmath.org/

Sequence

) of the

Padovan sequence

or the

Perrin sequence

, both of which obey the

recurrence relation

a

n

=

a

n

-

3

+

a

n

-

2

.

(1)

Supposing that such a limit

P

:=

lim

n

→

∞

⁡

a

n

+

1

a

n

(2)

exists (and is

≠

0

), we first write (1) as

a

n

a

n

-

1

⋅

a

n

-

1

a

n

-

2

=

a

n

-

3

a

n

-

2

+

1

(3)

and then let

n

→

∞

.  It follows the limit equation

P

⋅

P

=

1

P

+

1

,

which is same as

P

3

=

P

+

1

.

(4)

Thinking the graphs of the equations

y

=

x

3

and

y

=

x

+

1

, it is clear that the

cubic equation

x

3

-

x

-

1

=

0

(5)

has only one

real root

(

http://planetmath.org/Equation

), which is

P

.

For solving the plastic number from the cubic, substitute by Cardano (

http://planetmath.org/CardanosFormulae

) into (5) the sum

x

:=

u

+

v

of two auxiliary

unknowns

, when the equation may be written

(

u

3

+

v

3

-

1

)

+

(

3

⁢

u

⁢

v

-

1

)

⁢

(

u

+

v

)

=

0

.

Then, as in the example of solving a cubic equation,

u

and

v

are determined such that the first two parentheses vanish:

{

u

2

+

v

3

=

1

,

u

⁢

v

=

1

3

,

or

u

3

⁢

v

3

=

1

27

Thus

u

3

and

v

3

are the roots of the

resolvent equation

z

2

-

z

+

1

27

=

0

,

i.e.

z

=

9

±

69

18

and accordingly

u

=

9

+

69

18

3

,

v

=

9

-

69

18

3

.

Fixing that these the real values of the

cube roots

, we obtain the value of the plastic number in the form

x

=

9

+

69

18

3

+

9

-

69

18

3

,

or

P

=

12

⁢

(

9

+

69

)

3

+

12

⁢

(

9

-

69

)

3

6

.

(6)

By (5),

P

is an

algebraic integer

of degree (

http://planetmath.org/DegreeOfAnAlgebraicNumber

) 3 (and a unit of the

ring of integers

of the

number field

ℚ

⁢

(

P

)

).  For computing an approximate value of

P

, see e.g.

nth root

by Newton’s method.

Title

derivation of plastic number

Canonical name

DerivationOfPlasticNumber

Date of creation

2013-03-22 19:09:41

Last modified on

2013-03-22 19:09:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Derivation

Classification

msc 11B39

Related topic

LimitRulesOfSequences

Related topic

GoldenRatio

Related topic

LimitRulesOfFunctions

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

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Fibonacci fraction

Fibonacci fraction

A

Fibonacci fraction

is a

rational number

of the form

F

n

F

m

where

F

i

is the

i

th number of the

Fibonacci sequence

and

n

and

m

are

integers

in the

relation

n

<

m

. In the

Fibonacci

fractional series, each

m

=

n

+

2

:

1

2

,

1

3

,

2

5

,

3

8

,

5

13

,

8

21

,

13

34

,

21

55

,

34

89

,

…

The most important application of Fibonacci fractions is in botany: plants arrange the leaves on their stems (phyllotaxy) in many different ways, but “only those conforming to a Fibonacci fraction allow for efficient packing of leaf primordia on the meristem surface.” There is also an application in optics.

References

1

P. A. David “Leaf Position in Ailanthus Altissima in Relation to the Fibonacci Series”

American Journal of Botany

26

2 (1939): 67

2

R. W Pearcy & W Yang “The

functional

morphology of light capture and carbon gain in the Redwood forest understorey plant Adenocaulon bicolor Hook”

Functional Ecology

12

4 (1998): 551

3

H. C. Rosu, J. P. Trevino, H. Cabrera & J. S. Murguia, “Self-image effects for diffraction and dispersion”

Electromagnetic Phenomena

6

2 (2006): 204 - 211

Title

Fibonacci fraction

Canonical name

FibonacciFraction

Date of creation

2013-03-22 18:03:18

Last modified on

2013-03-22 18:03:18

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B39

Generated on Fri Feb 9 15:27:32 2018 by

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