aarjaneiro/planetmath huggingface dataset:
Feynman point

Feynman point

In the base 10 representation of

π

, at positions 762 through 767 (after the

decimal point

) there are six instances of the digit 9, called the

Feynman point

. Physicist

Richard Feynman

joked that he wanted to memorize

π

up to that point: 3.14159265358979323846264338 … 77130996051870721134999999… Such a recitation would erroneously suggest that

π

is a

rational number

, the

quotient

of two

integers

that happen to be

coprime

to the base.

Title

Feynman point

Canonical

name

FeynmanPoint

Date of creation

2013-03-22 16:26:01

Last modified on

2013-03-22 16:26:01

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:52 2018 by

LaTeXML
fixed-point arithmetic

fixed-point arithmetic

For computation or display in

fixed-point arithmetic

, each

real number

x

has a

variable

number

k

>

0

of

integer

digits

d

k

-

1

,

…

,

d

0

(in order left to right from the most to the least significant digit) in base

b

but always

l

fractional digits to the right of the point. Thus,

x

=

∑

i

=

-

l

k

-

1

d

i

⁢

b

i

.

For example, for the usual mundane applications of money,

b

=

10

and

l

=

2

.

Fixed-point arithmetic can adequately represent any

rational number

n

m

provided that

m

≤

b

l

and

m

has no

prime factors

b

doesn’t (counting repeated prime factors).

Addition

and

subtraction

in fixed-point arithmetic present no special problems for such numbers. When it comes to

multiplication

and

division

, we run into cases where the result cannot be adequately represented though the operands can.

In those cases it may be practical to perform the computations in floating point and display the results in fixed-point. In fact, this is what many business

calculators

do (such as cash registers); they compute in binary floating point and display in decimal fixed-point. Another option is to continue the calculations in fixed-point by settling for crude approximations. Depending on the application and the user, the loss of precision may be acceptable. There are two ways to handle the loss of precision: if

d

-

l

-

1

>

b

2

, we could then assign

d

-

l

=

d

-

l

+

1

(propagating the change as needed to the more significant digits); or we could simply ignore everything to the right of

d

-

l

.

Older versions of FractInt for 80386 IBM-compatibles used fixed-point arithmetic. Most

computers

(and even game consoles) today have floating point units, reducing the need for fixed-point. Even so, newer implementations of the C programming language are expected to provide fixed-point data types and appropriately

overload

built-in

functions

to handle those data types (see ISO/IEC TR 18037:2004).

Title

fixed-point arithmetic

Canonical

name

FixedpointArithmetic

Date of creation

2013-03-22 16:32:31

Last modified on

2013-03-22 16:32:31

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Synonym

fixed point arithmetic

Synonym

fixed point

Synonym

fixed-point

Generated on Fri Feb 9 15:22:54 2018 by

LaTeXML
Friedman number

Friedman number

Consider the

integer

28547. In the

equation

28547

=

(

8

+

5

)

4

-

(

7

×

2

)

expressed in base 10, both sides use the same digits.

An integer is a

Friedman number

if it can be put into an equation such that both sides use the same digits but the right hand side has one or more basic

arithmetic

operators (

addition

,

subtraction

,

multiplication

,

division

,

exponentiation

) interspersed. Brackets, as usual, are essential to clarify the

order of operations

. These numbers are named after Erich Friedman, Assoc. Professor of
Mathematics at Stetson University. With the help of his students he has researched Friedman numbers in bases 2 through 10 and even with Roman numerals.

When both sides use the digits in the same order, the number is called a ”nice” or ”strong” Friedman number. For example,

3125

=

(

3

+

[

1

×

2

]

)

5

.

These

concepts

can be transplanted into any standard positional numbering system using 0. Transplanting into the realm of Roman numerals, however, requires the addition of the extra constraint that the right hand side of the equation use at least one other operator besides addition or subtraction.

0.1

Links

http://www.stetson.edu/ efriedma/mathmagic/0800.html

Problem of the Month, August 2000

http://www.research.att.com/projects/OEIS?Anum=A036057

Sloane’s A036057

http://www.research.att.com/projects/OEIS?Anum=A080035

Sloane’s A080035

Title

Friedman number

Canonical

name

FriedmanNumber

Date of creation

2013-03-22 15:36:45

Last modified on

2013-03-22 15:36:45

Owner

aplant (12431)

Last modified by

aplant (12431)

Numerical id

10

Author

aplant (12431)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:22:55 2018 by

LaTeXML
frugal number

frugal number

A

frugal number

or

economical number

n

is an

integer

with a base

b

representation

of

k

digits for which the

prime factorization

uses fewer than

k

digits (with repeated

prime factors

grouped with

exponents

and the digits of those exponents counted whenever greater than 1). The first few frugal numbers in base 10 are 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250, 1280, 1331, 1369, 1458, 1536, 1681, 1701, 1715, 1792, 1849, 1875, etc. (listed in A046759 of Sloane’s OEIS). For example, 128 is frugal in base 10 because it is written with three digits, while its factorization of

2

7

uses just two digits. 128 also happens to be frugal in binary. If we regard 1 as not prime, then 1 is a frugal number in all positional bases (for example, Mathematica returns its factorization as an empty set).

References

1

D. Darling, “Economical number” in

The Universal Book of Mathematics: From Abracadabra To Zeno’s paradoxes

. Hoboken, New Jersey: Wiley (2004)

Title

frugal number

Canonical name

FrugalNumber

Date of creation

2013-03-22 16:41:15

Last modified on

2013-03-22 16:41:15

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Synonym

economical number

Related topic

EquidigitalNumber

Related topic

ExtravagantNumber

Generated on Fri Feb 9 15:22:57 2018 by

LaTeXML
happy number

happy number

Given a base

b

integer

n

=

∑

i

=

1

k

d

i

⁢

b

i

-

1

where

d

1

is the least significant digit and

d

k

is the most significant, and the function

f

⁢

(

m

)

=

∑

i

=

1

k

d

i

2

,

if computing

f

⁢

(

n

)

and iterating that function on the result

eventually

leads to a

fixed point

of 1, then

n

is said to be a

happy number

in base

b

.

For

b

=

2

and

b

=

4

, all numbers are happy numbers. All standard positional bases have happy numbers.

Title

happy number

Canonical name

HappyNumber

Date of creation

2013-03-22 16:19:56

Last modified on

2013-03-22 16:19:56

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:22:58 2018 by

LaTeXML
Harshad number

Harshad number

Consider the

integer

1729. Adding up its digits,

1

+

7

+

2

+

9

=

19

and

1729

19

=

91

.

When an integer is

divisible

by the sum of its digits, it’s called a

Harshad number

or

Niven number

. That is, given m is the number of digits of n and d is an integer of n,

∑

i

=

1

m

d

i

|

n

All 1-digit numbers and the base number itself are Harshad numbers. 1, 2, 4 and 6 are always Harshad numbers regardless of the base.

It is possible for an integer to be divisible by its

digital root

and yet not be a Harshad number because it doesn’t divide its first

digit sum

evenly (for example, 38 in base 10 has digital root 2 but is not divisible by 3 + 8 = 11). The reverse is also possible (for example, 195 is divisible by 1 + 9 + 5 = 15, but not by its digital root 4).

Title

Harshad number

Canonical name

HarshadNumber

Date of creation

2013-03-22 15:47:04

Last modified on

2013-03-22 15:47:04

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Synonym

Niven number

Defines

Harshad number

Generated on Fri Feb 9 15:23:00 2018 by

LaTeXML
hexadecimal

hexadecimal

The

hexadecimal

system

is a

positional number system

with base 16, using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.  It offers a compact way of expressing binary numbers.

In hexadecimal, all

Mersenne numbers

greater than 127 end with the digit F repeated several times, while all

Fermat numbers

greater than 17 are written with several significant zeroes book-ended by two 1’s.

The hexadecimal is a mixed formation of a Greek begin and a Latin end.  There is also a less used synonym

hexadecadic

of purely Greek .

Some

divisibility tests

in hexadecimal are:

n

is

divisible

by 2 if its least significant digit is 0, 2, 4, 6, 8, A, C or E.

n

is divisible by 4 if its least significant digit is 0, 4, 8 or C.

n

is divisible by 8 if its least significant digit is 0, or 8.

n

is divisible by 15 if it has

digital root

F.

n

is of course divisible by 16 if it ends in a 0.

n

is divisible by 17 if the

difference

of the odd placed digits and the even place digits of

n

is a

multiple

of 17.

Title

hexadecimal

Canonical

name

Hexadecimal

Date of creation

2013-03-22 16:20:27

Last modified on

2013-03-22 16:20:27

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Synonym

hexadecadic

Generated on Fri Feb 9 15:23:02 2018 by

LaTeXML
Kaprekar constant

Kaprekar constant

The

Kaprekar constant

K

k

in a given base

b

is a

k

-digit number

K

such that subjecting any other

k

-digit number

n

(except the

repunit

R

k

and numbers with

k

-

1

repeated digits) to the following process:

1. Arrange the digits of

n

in

ascending order

, forming the

k

-digit number

a

, and then in

descending order

, forming the

k

-digit number

b

.

2. If

a

>

b

, calculate

a

-

b

=

c

; otherwise

b

-

a

=

c

.

3. Goto step 1 using

c

instead of

n

.

eventually

gives

K

. (This process is sometimes called the

Kaprekar routine

).

For

b

=

10

, the Kaprekar constant for

k

=

4

is 6174. Using

n

=

1729

, we find that 9721 - 1279 gives 8442. Then 8442 - 2448 = 5994. Then 9954 - 4599 gives 5355. Then 5553 - 3555 gives 1998. Then 9981 - 1899 gives 8082. Then 8820 - 288 gives 8532. Then 8532 - 2538 finally gives 6174. (Some numbers take longer than others).

K

2

and

K

7

don’t exist for

b

=

10

.

Title

Kaprekar constant

Canonical name

KaprekarConstant

Date of creation

2013-03-22 16:16:30

Last modified on

2013-03-22 16:16:30

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Synonym

Kaprekar’s constant

Defines

Kaprekar routine

Generated on Fri Feb 9 15:23:03 2018 by

LaTeXML
Kaprekar number

Kaprekar number

Let

n

be a

k

-digit

integer

in base

b

. Then

n

is said to be a

Kaprekar number

in base

b

if

n

2

has the following property: when you add the number formed by its right hand digits to that formed by its left hand digits, you get

n

.

Or to put it algebraically, an integer

n

such that in a given base

b

has

n

2

=

∑

i

=

0

k

-

1

d

i

⁢

b

i

(where

d

x

are digits, with

d

0

the least

significant digit

and

d

k

-

1

the most significant) such that

∑

i

=

k

2

+

1

k

d

i

⁢

b

i

-

k

2

-

1

+

∑

i

=

1

k

2

d

i

⁢

b

i

-

1

=

n

if

k

is even or

∑

i

=

⌈

k

2

⌉

k

d

i

⁢

b

i

-

⌊

k

2

⌋

-

1

+

∑

i

=

1

k

2

d

i

⁢

b

i

-

1

=

n

if

k

is odd.

b

x

-

1

for a natural

x

is always a Kaprekar number in base

b

.

References

1

D. R. Kaprekar, “On Kaprekar numbers”

J. Rec. Math.

13 (1980-1981), 81 - 82.

Title

Kaprekar number

Canonical name

KaprekarNumber

Date of creation

2013-03-22 16:00:17

Last modified on

2013-03-22 16:00:17

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:05 2018 by

LaTeXML
Keith number

Keith number

Given a base

b

integer

n

=

∑

i

=

1

k

d

i

⁢

b

i

-

1

where

d

1

is the least

significant digit

and

d

k

is the most significant, construct the

sequence

a

1

=

d

k

,

…

⁢

a

k

=

d

1

, and for

m

>

k

,

a

m

=

∑

i

=

1

k

a

m

-

i

.

If there is an

x

such that

a

x

=

n

, then

n

is a

Keith number

or

repfigit number

.

In base 10, the first few Keith numbers below ten thousand are: 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909 (see A007629 in Sloane’s OEIS for a longer listing). 47 is a base 10 Keith number because it is contained the Fibonacci-like recurrence started from its base 10 digits: 4, 7, 11, 18, 29, 47, etc.

References

1

M. Keith, “Repfigit Numbers”

J. Rec. Math.

19 (1987), 41 - 42.

Title

Keith number

Canonical name

KeithNumber

Date of creation

2013-03-22 16:00:20

Last modified on

2013-03-22 16:00:20

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Synonym

repfigit number

Generated on Fri Feb 9 15:23:07 2018 by

LaTeXML
Kummer’s theorem

Kummer’s theorem

Given

integers

n

≥

m

≥

0

and a

prime number

p

, then the power of

p

dividing

(

n

m

)

is equal to the number of carries when adding

m

and

n

-

m

in base

p

.

Proof.

For the proof we can allow of numbers in base

p

with leading
zeros. So let

n

d

⁢

n

d

-

1

⁢

⋯

⁢

n

0

:=

n

,

m

d

⁢

m

d

-

1

⁢

⋯

⁢

m

0

:=

m

,

all in base

p

. We set

r

=

n

-

m

and denote the

p

-adic

representation

of

r

with

r

d

⁢

r

d

-

1

⁢

⋯

⁢

r

0

.

We define

c

-

1

=

0

, and for each

0

≤

j

≤

d

c

j

=

{

1

for

m

j

+

r

j

≥

p

0

otherwise.

(1)

Finally, we introduce

δ

p

⁢

(

n

)

as the sum of digits in the

p

-adic
of

n

. Then it follows that the power of

p

dividing

(

n

m

)

is

δ

p

⁢

(

m

)

+

δ

p

⁢

(

r

)

-

δ

p

⁢

(

n

)

p

-

1

.

For each

j

≥

0

, we have

n

j

=

m

j

+

r

j

+

c

j

-

1

-

p

.

c

j

.

Then

δ

p

⁢

(

m

)

+

δ

p

⁢

(

r

)

-

δ

p

⁢

(

n

)

=

∑

k

=

0

d

(

m

k

+

r

k

-

n

k

)

=

∑

k

=

0

d

(

(

p

-

1

)

⁢

c

j

)

+

∑

k

=

0

d

(

c

j

-

c

j

-

1

)

=

∑

k

=

0

d

(

p

-

1

)

⁢

c

j

+

c

d

-

c

-

1

This gives us

δ

p

⁢

(

m

)

+

δ

p

⁢

(

r

)

-

δ

p

⁢

(

n

)

p

-

1

=

∑

k

=

0

d

c

k

,

the total number of carries.
∎

Title

Kummer’s theorem

Canonical name

KummersTheorem

Date of creation

2013-03-22 13:22:37

Last modified on

2013-03-22 13:22:37

Owner

Thomas Heye (1234)

Last modified by

Thomas Heye (1234)

Numerical id

14

Author

Thomas Heye (1234)

Entry type

Theorem

Classification

msc 11A63

Generated on Fri Feb 9 15:23:09 2018 by

LaTeXML
least significant digit

least significant digit

The

least

significant digit

of a number

n

written in a given positional base

b

is the digit in the least significant place value, and has to be in the range

-

1

<

d

1

<

b

. In the case of an

integer

, the least significant digit is the 1’s place value, usually written to the right of the

b

’s place value. In the case of a

transcendental number

, there is no actual least significant digit, but for computational purposes the

rational approximation

would have a least significant digit.

In an array of digits

k

long meant for mathematical manipulation, it might be convenient to index the least significant digit with index 1 or 0, and the more significant digits with larger integers. (This enables the calculation of the value of a given digit as

d

i

⁢

b

i

rather than

d

i

⁢

b

k

-

i

.) For an array of digits meant for text string manipulation, however, the least significant digit might be placed at position

k

(for example, by Mathematica’s IntegerDigits function).

In binary, the least significant digit is often called the

least significant bit

.

Title

least significant digit

Canonical name

LeastSignificantDigit

Date of creation

2013-03-22 16:21:06

Last modified on

2013-03-22 16:21:06

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Defines

least significant bit

Generated on Fri Feb 9 15:23:11 2018 by

LaTeXML
Leyland number

Leyland number

An

integer

of the form

x

y

+

y

x

, with

1

<

x

≤

y

.

Leyland numbers

with

x

<

y

<

11

are

8, 17, 32, 57, 100, 177, 320, 593, 1124
54, 145, 368, 945, 2530, 7073, 20412, 60049
512, 1649, 5392, 18785, 69632, 268705, 1058576
6250, 23401, 94932, 423393, 2012174, 9865625
93312, 397585, 1941760, 10609137, 61466176
1647086, 7861953, 45136576, 292475249
33554432, 177264449, 1173741824
774840978, 4486784401
20000000000

The largest known prime Leyland number

2638

4405

+

4405

2638

.

Reference

R. Crandall, C. Pomerance,

Prime Numbers

: A Computational Perspective, Springer, 2001

Title

Leyland number

Canonical

name

LeylandNumber

Date of creation

2013-03-22 15:50:01

Last modified on

2013-03-22 15:50:01

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:13 2018 by

LaTeXML
look and say sequence

look and say sequence

The

look and say sequence

a

for a given integer

n

in base

b

consists of each

a

i

being the

run-length encoding

of the digits of

a

i

-

1

for

i

>

0

, with

a

0

=

n

.

For example, with

n

=

7777

and

b

=

10

, the number consists of four 7s, so its run-length encoding is 47 and

a

1

=

47

. Then in turn, 47 is one 4 and one 7, so

a

2

=

1417

. The sequence continues 11141117, 31143117, 132114132117, etc.

John Horton Conway has thoroughly studied look and say sequences in base 10 starting with values containing only 1s, 2s and 3s and no more than three

consecutive

repetitions of any digit. With the exception of 22 (two 2s), all other such starting values

eventually

“decay” to one of 92 “atoms” which Conway has named after the elements of the periodic table up to uranium. The whole

concept

he has termed with the awful pun “audioactive decay.”

The base 10 look and say sequences starting with 1, 2 and 3 are given in A005150, A006751 and A006715 respectively in Sloane’s OEIS.

References

1

Steven R. Finch,

Mathematical Constants

. Cambridge: Cambridge University Press (2003): 452 - 455

Title

look and say sequence

Canonical name

LookAndSaySequence

Date of creation

2013-03-22 18:02:34

Last modified on

2013-03-22 18:02:34

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:14 2018 by

LaTeXML
most significant digit

most significant digit

The

most

significant digit

of a number

n

written in a given positional base

b

is the digit in the most significant place value, and has to be in the range

-

1

<

d

k

<

b

. In the case of an integer, the most significant digit is the

b

k

’s place value, where

k

is the total number of digits, or

k

=

⌊

l

⁢

o

⁢

g

b

⁢

n

⌋

.

In an array of digits

k

long meant for mathematical manipulation, it might be convenient to index the least significant digit with index 1 or 0, and the more significant digits with larger integers. (This enables the calculation of the value of a given digit as

d

i

⁢

b

i

rather than

d

i

⁢

b

k

-

i

.) For an array of digits meant for text string manipulation, however, the most significant digit might be placed at position 0 or 1 (for example, by Mathematica’s IntegerDigits function).

In binary, the most significant digit is often called the

most significant bit

.

Title

most significant digit

Canonical

name

MostSignificantDigit

Date of creation

2013-03-22 16:52:20

Last modified on

2013-03-22 16:52:20

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:16 2018 by

LaTeXML
multiplicative digital root

multiplicative digital root

Given an

integer

m

consisting of

k

digits

d

1

,

…

,

d

k

in base

b

, let

j

=

∏

i

=

1

k

d

i

,

then repeat this

operation

on the digits of

j

until

j

<

b

. This stores in

j

the

multiplicative digital root

of

m

. The number of iterations of the

multiplication

operation is called the

multiplicative persistence

of

m

.

Title

multiplicative digital root

Canonical

name

MultiplicativeDigitalRoot

Date of creation

2013-03-22 16:00:42

Last modified on

2013-03-22 16:00:42

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Synonym

repeated digit product

Synonym

repeated digital

product

Defines

multiplicative persistence

Generated on Fri Feb 9 15:23:18 2018 by

LaTeXML
palindromic number

palindromic number

An

integer

n

that in a given base

b

is a

palindrome

. Given

n

being

k

digits

d

x

long in base

b

, its value being

n

=

∑

i

=

0

k

-

1

d

k

⁢

b

i

then if the equalities

d

k

=

d

1

,

d

k

-

1

=

d

2

, etc., hold, then

n

is a

palindromic number

. There are infinitely many palindromic numbers in any given base.

Title

palindromic number

Canonical name

PalindromicNumber

Date of creation

2013-03-22 15:55:27

Last modified on

2013-03-22 15:55:27

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:19 2018 by

LaTeXML
palindromic prime

palindromic prime

A

prime number

p

that in a given base

b

is also a

palindromic number

. Although there are infinitely many palindromic numbers in any given base, it is not known if the same is true of

palindromic primes

.

If

b

+

1

is prime, then it is the only palindromic prime in base

b

to have an

even number

of digits; all other

palindromes

with an even number of digits will be

multiples

of

b

+

1

.

Title

palindromic prime

Canonical name

PalindromicPrime

Date of creation

2013-03-22 15:55:30

Last modified on

2013-03-22 15:55:30

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:21 2018 by

LaTeXML