aarjaneiro/planetmath huggingface dataset:
result on quadratic residues

result on quadratic residues

Theorem.

Let

p

be an

odd prime

. Then

-

3

is a

quadratic residue

modulo

p

if and only if

p

≡

1

(

mod

3

)

.

Proof.

Preliminary to the proof, we remark first that

-

1

is a quadratic residue modulo

p

, where

p

is an odd prime, if and only if

p

≡

1

(

mod

4

)

.

If

p

≡

1

(

mod

4

)

then

(

-

3

p

)

=

(

-

1

p

)

⁢

(

3

p

)

=

(

p

3

)

.

Now if

p

≡

3

(

mod

4

)

, then

(

-

3

p

)

=

(

-

1

p

)

⁢

(

3

p

)

=

(

-

1

)

⁢

(

-

1

)

⁢

(

p

3

)

=

(

p

3

)

.

Thus,

(

-

3

p

)

=

(

p

3

)

, and

(

p

3

)

=

1

if and only if

p

≡

1

(

mod

3

)

.

□

Title

result on quadratic residues

Canonical name

ResultOnQuadraticResidues

Date of creation

2013-03-22 16:08:09

Last modified on

2013-03-22 16:08:09

Owner

gilbert_51126 (14238)

Last modified by

gilbert_51126 (14238)

Numerical id

25

Author

gilbert_51126 (14238)

Entry type

Theorem

Classification

msc 11-00

Generated on Fri Feb 9 15:08:32 2018 by

LaTeXML
square

square

The

square

of a number

x

is the number obtained multiplying

x

by itself.  It’s denoted as

x

2

.

Some examples:

5

2

=

25

(

1

3

)

2

=

1

9

0

2

=

0

(

-

0.5

)

2

=

0.25

A

perfect square

is the square of an

integer

. The first few perfect squares are 0, 1, 4, 9, 16, 25,

…

.

Title

square

Canonical name

Square1

Date of creation

2013-03-22 12:02:35

Last modified on

2013-03-22 12:02:35

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Definition

Classification

msc 11-00

Classification

msc 26-00

Classification

msc 30-00

Synonym

second power

Related topic

CubeOfANumber

Related topic

PerfectPower

Defines

perfect square

Generated on Fri Feb 9 15:08:36 2018 by

LaTeXML
the difference of two odd squares is always a multiple of 8

the difference of two odd squares is always a multiple of 8

Theorem

.

Given any odd real

integers

m

and

n

, the

difference

of their squares is always a

multiple

of 8. That is,

8

|

(

m

2

-

n

2

)

.

For example, given

m

=

9

and

n

varied from

-

1

to 13 in steps of 2, calculating

m

2

-

n

2

, we get the

sequence

80, 80, 72, 56, 32, 0,

-

40

,

-

88

.

Proving this theorem was one of the problems in the 1851 math exam Lewis Carroll took. The difficulty of this problem is one of perception: so often a simple-sounding theorem in

number theory

turns out to have a rather complicated proof. But for this simple theorem, the proof turns out to be quite simple, almost bordering on triviality.

Proof.

Since it was stipulated that

m

and

n

are both odd, we can rewrite them as

m

=

2

⁢

a

+

1

and

n

=

2

⁢

b

+

1

. The square of

m

is then

m

2

=

(

2

⁢

a

+

1

)

⁢

(

2

⁢

a

+

1

)

=

4

⁢

a

2

+

2

⁢

a

+

2

⁢

a

+

1

, which simplifies to

4

⁢

a

2

+

4

⁢

a

+

1

. Likewise,

n

2

=

4

⁢

b

2

+

4

⁢

b

+

1

. Their difference is then

(

4

⁢

a

2

+

4

⁢

a

+

1

)

-

(

4

⁢

b

2

+

4

⁢

b

+

1

)

=

(

4

⁢

a

2

+

4

⁢

a

)

-

(

4

⁢

b

2

+

4

⁢

b

)

. This can’t be reduced to fewer terms. However, either side can be rewritten so as to not explicitly use squaring:

4

⁢

a

2

+

4

⁢

a

=

4

⁢

a

⁢

(

a

+

1

)

. This reveals that either the left-hand or right-hand side of our

subtraction

is a number that is four times a

pronic number

, that is, a number of the form

a

⁢

(

a

+

1

)

. All pronic numbers are even, but at this point we can’t distinguish between

singly even numbers

and

doubly even numbers

. But, as it turns out, each pronic number is twice a

triangular number

, which is an integer. So

4

⁢

a

⁢

(

a

+

1

)

is eight times some triangular number

T

c

, and

4

⁢

b

2

+

4

⁢

b

=

8

⁢

T

d

. We rewrite our subtraction yet again:

m

2

-

n

2

=

8

⁢

T

c

-

8

⁢

T

d

. By redistributing, we get

m

2

-

n

2

=

8

⁢

(

T

c

-

T

d

)

, proving the theorem.
∎

The exam paper which Lewis Carroll turned in has not survived, so we don’t know in what way he solved this particular problem or any of the other problems on that test. But we do know that he passed the test with flying colors.

Title

the difference of two odd squares is always a multiple of 8

Canonical

name

TheDifferenceOfTwoOddSquaresIsAlwaysAMultipleOf8

Date of creation

2013-03-22 18:47:06

Last modified on

2013-03-22 18:47:06

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 11-00

Classification

msc 30-00

Classification

msc 26-00

Generated on Fri Feb 9 15:08:41 2018 by

LaTeXML
using primitive roots and index to solve congruences

using primitive roots and index to solve congruences

The aim of the following example is to illustrate how to use

primitive roots

and the index of an

integer

to solve seemingly complicated

congruences

.

For this example, let

p

=

13

and let us attempt to solve

7

⁢

x

10

≡

5

mod

13

.

Since every prime has a primitive root, we can easily find one. In particular, the number

2

is a primitive root for

p

=

13

. Indeed, the powers of

2

are the following modulo

13

:

x

x

2

x

3

x

4

x

5

x

6

x

7

x

8

x

9

x

10

x

11

x

12

2

4

8

3

6

12

11

9

5

10

7

1

The table above allows us to build a table of indices with base

2

(for the definition of index and its properties which will be used below, see

http://planetmath.org/node/PropertiesOfTheIndexOfAnIntegerWithRespectToAPrimitiveRoot

this entry):

x

1

2

3

4

5

6

7

8

9

10

11

12

ind

⁡

x

12

1

4

2

9

5

11

3

8

10

7

6

Now we can use the properties of index to solve the equation

7

⁢

x

10

≡

5

mod

13

. By taking indices on both sides we obtain

ind

⁡

(

7

⁢

x

10

)

≡

ind

⁡

7

+

10

⁢

ind

⁡

x

≡

ind

⁡

5

mod

12

and so

ind

⁡

x

≡

ind

⁡

5

-

ind

⁡

7

≡

9

-

11

≡

10

mod

12

. The

equivalence

10

⁢

ind

⁡

x

≡

10

mod

12

implies

5

⁢

ind

⁡

x

≡

5

mod

6

and hence

ind

⁡

x

≡

1

mod

6

.

Lifting

this solution to modulo

12

we obtain

ind

⁡

x

≡

1

or

7

modulo

12

, and by the table of indices,

x

≡

2

or

11

modulo

13

are the unique solutions of the congruence.

Title

using primitive roots and index to solve congruences

Canonical name

UsingPrimitiveRootsAndIndexToSolveCongruences

Date of creation

2013-03-22 16:20:58

Last modified on

2013-03-22 16:20:58

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Example

Classification

msc 11-00

Related topic

PrimitiveRoot

Related topic

Congruence

Related topic

PolynomialCongruence

Generated on Fri Feb 9 15:08:43 2018 by

LaTeXML
values of the Legendre symbol

values of the Legendre symbol

For an

integer

a

and an

odd prime

p

, let

(

a

p

)

be the

Legendre symbol

.

Theorem

.

Let

p

be an odd prime. The Legendre symbol takes the following values:

1.

(

-

1

p

)

=

{

1

if

⁢

p

≡

1

mod

4

-

1

if

⁢

p

≡

3

mod

4

.

2.

(

2

p

)

=

{

1

if

⁢

p

≡

±

1

mod

8

-

1

if

⁢

p

≡

3

,

5

mod

8

.

3.

(

3

p

)

=

{

1

if

⁢

p

≡

±

1

mod

12

-

1

otherwise.

4.

(

5

p

)

=

{

1

if

⁢

p

≡

±

1

mod

5

-

1

if

⁢

p

≡

2

,

3

mod

5

.

Proof.

For a proof of (1), see

http://planetmath.org/node/1IsQuadraticResidueIfAndOnlyIfPequiv1Mod4

this entry. Part (2) is proved in

http://planetmath.org/node/QuadraticCharacterOf2

this entry. For parts (3), (4) and (5), we use

quadratic reciprocity

. For example,

(

5

p

)

=

(

p

5

)

and the only

quadratic residues

modulo

5

are

±

1

mod

5

.
∎

Title

values of the Legendre symbol

Canonical name

ValuesOfTheLegendreSymbol

Date of creation

2013-03-22 16:18:13

Last modified on

2013-03-22 16:18:13

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11-00

Related topic

1IsQuadraticResidueIfAndOnlyIfPequiv1Mod4

Related topic

QuadraticCharacterOf2

Generated on Fri Feb 9 15:08:45 2018 by

LaTeXML
Wilson’s theorem result

Wilson’s theorem result

The following is a variant of Wilson’s theorem:

Let

p

be a

positive integer

and

1

≤

k

≤

p

-

1

. Then:

p

is prime if and only if

(

p

-

k

)

!

⁢

(

k

-

1

)

!

≡

(

-

1

)

k

(

mod

p

)

.

For particular

k

=

1

,

2

,

…

one gets nice results.

Title

Wilson’s theorem result

Canonical name

WilsonsTheoremResult

Date of creation

2013-03-22 14:20:44

Last modified on

2013-03-22 14:20:44

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Result

Classification

msc 11-00

Generated on Fri Feb 9 15:08:46 2018 by

LaTeXML
lcm⁢(m⁢a,m⁢b)=m⁢lcm⁢(a,b)

lcm

⁢

(

m

⁢

a

,

m

⁢

b

)

=

m

⁢

lcm

⁢

(

a

,

b

)

For simplicity, let us work only with

positive integers

.

We want to prove that if a,b,m are

integers

, then

lcm

⁢

(

m

⁢

a

,

m

⁢

b

)

=

m

⁢

lcm

⁢

(

a

,

b

)

.

First notice that any common

multiple

of

m

⁢

a

and

m

⁢

b

is also a multiple of

m

, so any common multiple of

m

⁢

a

and

m

⁢

b

is of the form

m

⁢

k

with some integer

k

.

Now notice that if

t

=

lcm

⁢

(

a

,

b

)

and

u

<

t

, it cannot happen that

a

∣

u

and

b

∣

u

, since

t

is the smallest number, So, when

a

∤

u

then

m

⁢

a

∤

m

⁢

u

, and if

b

∤

u

then

m

⁢

b

∤

m

⁢

u

. We conclude that

m

⁢

u

is

not

a common multiple of

m

⁢

a

and

m

⁢

b

when

u

<

t

.

So far, we proved that

m

⁢

t

=

m

⁢

lcm

⁢

(

a

,

b

)

is a common multiple of

m

⁢

a

and

m

⁢

b

, and previous paragraph shows that there is no smaller common multiple, therefore

m

⁢

lcm

⁢

(

a

,

b

)

is the

least

common multiple of

m

⁢

a

and

m

⁢

b

, in other words:

lcm

⁢

(

m

⁢

a

,

m

⁢

b

)

=

m

⁢

lcm

⁢

(

a

,

b

)

.

Title

lcm

⁢

(

m

⁢

a

,

m

⁢

b

)

=

m

⁢

lcm

⁢

(

a

,

b

)

Canonical name

mathrmlcmmambmmathrmlcmab

Date of creation

2013-03-22 15:03:22

Last modified on

2013-03-22 15:03:22

Owner

drini (3)

Last modified by

drini (3)

Numerical id

12

Author

drini (3)

Entry type

Theorem

Classification

msc 11-00

Generated on Fri Feb 9 15:08:48 2018 by

LaTeXML
fraction

fraction

A

fraction

is a

rational number

expressed in the form

n

d

or

n

/

d

, where

n

is designated the

numerator

and

d

the

denominator

. The slash between them is known as a

solidus

when the fraction is expressed as

n

/

d

.

The fraction

n

/

d

has value

n

÷

d

. For

instance

,

3

/

2

=

3

÷

2

=

1.5

.

If

n

and

d

are

positive

, and

n

/

d

<

1

, then

n

/

d

is known as a

proper fraction

. Otherwise, it is an

improper fraction

. If

n

and

d

are relatively prime, then

n

/

d

is said to be in

lowest terms

. Each rational number can be expressed uniquely as a fraction in lowest terms. To get a fraction in lowest terms, simply divide the numerator and the denominator by their

greatest common divisor

:

60

84

=

60

÷

12

84

÷

12

=

5

7

.

The rules for manipulating fractions are

a

b

=

k

⁢

a

k

⁢

b

a

b

+

c

d

=

a

⁢

d

+

b

⁢

c

b

⁢

d

a

b

-

c

d

=

a

⁢

d

-

b

⁢

c

b

⁢

d

a

b

×

c

d

=

a

⁢

c

b

⁢

d

a

b

÷

c

d

=

a

⁢

d

b

⁢

c

.

Title

fraction

Canonical name

Fraction

Date of creation

2013-03-22 12:34:11

Last modified on

2013-03-22 12:34:11

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

11

Author

bwebste (988)

Entry type

Definition

Classification

msc 11-01

Related topic

RationalNumber

Related topic

Number

Related topic

CategoryOfAdditiveFractions

Defines

solidus

Defines

proper fraction

Defines

numerator

Defines

denominator

Defines

improper fraction

Defines

lowest terms

Generated on Fri Feb 9 15:09:03 2018 by

LaTeXML
mixed fraction

mixed fraction

Any

improper fraction

a

b

can be uniquely written as a sum of an

integer

q

and a proper fraction

r

b

:

a

b

=

q

+

r

b

(1)

The decomposition (1) is guaranteed by the

division algorithm for integers

.  The sum form is called

mixed fraction

or

mixed number

.

For explicitly given integers

q

,

r

,

b

, the mixed fraction is usually written without the plus sign, for example

9

2

=

4

⁤

1

2

.

Given an improper fraction of the

positive integers

a

and

b

, their

division

gives the incomplete quotient

q

and the

remainder

r

; then the mixed fraction can be gotten by (1).

On the other hand, if one wants to convert e.g.

8

⁤

11

15

into an improper fraction, one needs first to convert the

integer part

8 into fifteenths, obtaining

8

=

8

⋅

15

15

=

120

15

,

this fraction is added to the ready fifteenths

11

15

; the total amount of fifteenths is

131

15

; thus one has

8

⁤

11

15

=

131

15

.

Note.

A minus sign in front of a mixed fraction belongs not only for the integer part but also for the

fractional part

, i.e. for example

-

2

⁤

1

4

=

-

2

-

1

4

.

The

rational expressions

, i.e. denoted quotients of

polynomials

, have analogous to mixed fractions.  E.g., if the degree of the polynomial

A

⁢

(

x

)

is at least equal to the degree of

B

⁢

(

x

)

, we have

A

⁢

(

x

)

B

⁢

(

x

)

=

Q

⁢

(

x

)

+

R

⁢

(

x

)

B

⁢

(

x

)

.

(2)

Here the degree of the remainder polynomial

R

⁢

(

x

)

is less than the degree of the numerator polynomial

B

⁢

(

x

)

, an arithmetic fact that has been well documented for over 4,000 years.

Fibonacci

in the

Liber Abaci

used mixed fraction arithmetic in 1202 AD.

Archimedes Calculus

included mixed fractions to record the area of a parabola. Egyptian scribes in 1650 BCE also used this class of mixed fraction arithmetic. One scribe, Ahmes scaled a volume unit in

multiples

of 64 so that a quotient recorded 1/64 units and a remainder recorded 1/320 units.

Title

mixed fraction

Canonical name

MixedFraction

Date of creation

2013-03-22 19:18:57

Last modified on

2013-03-22 19:18:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 11-01

Synonym

mixed number

Related topic

RationalNumber

Related topic

LongDivision

Related topic

EuclideanNumberField

Related topic

PartialFractionsOfExpressions

Generated on Fri Feb 9 15:09:04 2018 by

LaTeXML
Euclid’s lemma

Euclid’s lemma

If

a

,

b

, and

c

are

integers

(or, more generally, elements of a

principal ideal domain

) with

a

|

b

c

and

gcd

⁢

(

a

,

b

)

=

1

, then

a

|

c

.

Title

Euclid’s lemma

Canonical name

EuclidsLemma

Date of creation

2013-03-22 11:46:53

Last modified on

2013-03-22 11:46:53

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

9

Author

KimJ (5)

Entry type

Theorem

Classification

msc 11-02

Classification

msc 11A05

Generated on Fri Feb 9 15:09:08 2018 by

LaTeXML
a formula for amicable pairs

a formula for amicable pairs

The following

formula

is due to

Thabit ibn Qurra

(836-901), a mathematician who worked in Baghdad’s “House of Wisdom” translating Greek and Syrian works (such as Apollonius’s “Conics” or works of Euclid and Archimedes). As he translated the texts, ibn Qurra produced a mathematical body of his own.

Theorem

.

Let

n

≥

1

be a

natural number

and suppose that the numbers

3

⋅

2

n

-

1

,

3

⋅

2

n

-

1

-

1

and

9

⋅

2

2

⁢

n

-

1

-

1

are all prime. Then the numbers:

2

n

⁢

(

3

⋅

2

n

-

1

)

⁢

(

3

⋅

2

n

-

1

-

1

)

and

2

n

⁢

(

9

⋅

2

2

⁢

n

-

1

-

1

)

are

amicable numbers

.

Example

.

When

n

=

2

one has:

3

⋅

2

2

-

1

=

11

,

3

⋅

2

2

-

1

-

1

=

5

and

9

⋅

2

4

-

1

-

1

=

71

which are all primes. Thus, the numbers:

2

2

⁢

(

3

⋅

2

2

-

1

)

⁢

(

3

⋅

2

2

-

1

-

1

)

=

220

and

2

2

⁢

(

9

⋅

2

4

-

1

-

1

)

=

284

form an amicable pair. In fact, this is the smallest amicable pair. For

n

=

4

one obtains the amicable pair

17296

and

18416

.

Title

a

formula for amicable pairs

Canonical name

AFormulaForAmicablePairs

Date of creation

2013-03-22 15:52:45

Last modified on

2013-03-22 15:52:45

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 11A05

Related topic

ThabitNumber

Generated on Fri Feb 9 15:09:10 2018 by

LaTeXML
abundance

abundance

Given an

integer

n

with

divisors

d

1

,

…

,

d

k

(where the divisors are in

ascending order

and

d

1

=

1

,

d

k

=

n

) the

difference

(

∑

i

=

1

k

d

i

)

-

2

⁢

n

is the

abundance

of

n

. Or if one prefers,

(

∑

i

=

1

k

-

1

d

i

)

-

n

.

For example, the divisors of 12 (which are 1, 2, 3, 4, 6 and 12) add up to 28, which is 4 more than 24 (twice 12). Therefore, 12 has an abundance of 4. For the sake of comparison, the divisors of 13 are 1 and 13, adding up to 14, which is 12 less than 26 (twice 13). Therefore, 13 has an abundance of

-

12

. A033880 in Sloane’s OEIS lists the abundance of the first sixty-three

positive integers

.

Thus numbers with

positive

abundance are

abundant numbers

. A number with an abundance of exactly 1 is a

quasiperfect number

, while a number with 0 abundance is a

perfect number

. A number with an abundance of

-

1

is an almost perfect number (this is true of all powers of 2); all numbers with negative abundance are

deficient numbers

.

Title

abundance

Canonical

name

Abundance

Date of creation

2013-03-22 16:05:49

Last modified on

2013-03-22 16:05:49

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

9

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A05

Related topic

Deficiency

Generated on Fri Feb 9 15:09:11 2018 by

LaTeXML
abundant number

abundant number

An

integer

n

is an

abundant number

if the sum of the

proper divisors

of

n

is more than

n

itself, or the sum of all the

divisors

is more than twice

n

. That is,

σ

⁢

(

n

)

>

2

⁢

n

, with

σ

⁢

(

n

)

being the

sum of divisors function

.

For example, the integer 30. Its proper divisors are 1, 2, 3, 5, 6, 10, 15, which add up to 42.

Multiplying a

perfect number

by some integer

x

gives an abundant number (as long as

x

>

1

).

Given a pair of

amicable numbers

, the lesser of the two is abundant, its proper divisors adding up to the greater of the two.

Title

abundant number

Canonical name

AbundantNumber

Date of creation

2013-03-22 15:52:21

Last modified on

2013-03-22 15:52:21

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

6

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A05

Related topic

AmicableNumbers

Generated on Fri Feb 9 15:09:13 2018 by

LaTeXML
almost perfect number

almost perfect number

An

almost

perfect number

or

least

deficient number

is a number

n

whose

proper divisors

add up to just one less than itself. That is,

σ

⁢

(

n

)

-

n

=

n

-

1

, with

σ

⁢

(

n

)

being the

sum of divisors function

. Currently, the only known almost perfect numbers are the

integer

powers of 2 (e.g., 1, 2, 4, 8, 16, 32, 64, 128, etc.) No one has been able to prove that there are almost perfect numbers of a different form.

Title

almost perfect number

Canonical name

AlmostPerfectNumber

Date of creation

2013-03-22 17:41:51

Last modified on

2013-03-22 17:41:51

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A05

Synonym

least deficient number

Related topic

QuasiperfectNumber

Generated on Fri Feb 9 15:09:14 2018 by

LaTeXML
alternative proof of Euclid’s lemma

alternative proof of Euclid’s lemma

We give an alternative proof (see Euclid’s lemma proof), which does not use the

Fundamental Theorem of Arithmetic

(since, usually, Euclid’s lemma is used to prove FTA).

Lemma 1

.

If

a

∣

b

c

and

gcd

⁡

(

a

,

b

)

=

1

then

a

∣

c

.

Proof.

By

assumption

gcd

⁡

(

a

,

b

)

=

1

, thus we can use Bezout’s lemma to find

integers

x

,

y

such that

a

⁢

x

+

b

⁢

y

=

1

. Hence

c

⋅

(

a

⁢

x

+

b

⁢

y

)

=

c

and

a

⁢

c

⁢

x

+

b

⁢

c

⁢

y

=

c

. Since

a

∣

a

and

a

∣

b

c

(by

hypothesis

), we conclude that

a

∣

a

c

x

+

b

c

y

=

c

as claimed.
∎

Title

alternative proof of Euclid’s lemma

Canonical name

AlternativeProofOfEuclidsLemma

Date of creation

2013-03-22 14:12:27

Last modified on

2013-03-22 14:12:27

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Proof

Classification

msc 11A05

Generated on Fri Feb 9 15:09:16 2018 by

LaTeXML
amicable numbers

amicable numbers

Definition

.

Two

natural numbers

n

and

m

are

amicable numbers

(an amicable pair or simply amicable) if each is the sum of the

proper divisors

of the other.

Example

.

The smallest amicable pair is

n

=

220

and

m

=

284

. Indeed the sum of proper divisors of

220

is:

1

+

2

+

4

+

5

+

10

+

11

+

20

+

22

+

44

+

55

+

110

=

284

and the sum of proper divisors of

284

is:

1

+

2

+

4

+

71

+

142

=

220

.

Title

amicable numbers

Canonical name

AmicableNumbers

Date of creation

2013-03-22 15:52:41

Last modified on

2013-03-22 15:52:41

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 11A05

Synonym

amicable pair

Synonym

amicable

Related topic

AbundantNumber

Related topic

DeficientNumber

Generated on Fri Feb 9 15:09:17 2018 by

LaTeXML