aarjaneiro/planetmath huggingface dataset:
antiharmonic number

antiharmonic number

The antiharmonic, a.k.a.

contraharmonic mean

of some set of
positive numbers is defined as the sum of their squares
divided by their sum.  There exist

positive integers

n

whose sum

σ

1

⁢

(

n

)

of all their

positive

divisors

divides
the sum

σ

2

⁢

(

n

)

of the squares of those divisors.  For
example, 4 is such an

integer

:

1

+

2

+

4

=

7

∣

21

=

1

2

+

2

2

+

4

2

Such integers are called

antiharmonic numbers

(or

contraharmonic numbers

),
since the contraharmonic mean of their positive divisors is an
integer.

The antiharmonic numbers form the

HTTP://oeis.org/

OEIS

integer sequence

http://oeis.org/search?q=A020487&language=english&go=Search

A020487:

1

,

4

,

9

,

16

,

20

,

25

,

36

,

49

,

50

,

64

,

81

,

100

,

117

,

121

,

144

,

169

,

180

,

…

Using the expressions of

divisor function

(

http://planetmath.org/DivisorFunction

)

σ

z

⁢

(

n

)

, the condition for an
integer

n

to be an antiharmonic number, is that the

quotient

σ

2

⁢

(

n

)

:

σ

1

⁢

(

n

)

=

∑

0

⁢

<

d

∣

⁢

n

d

2

:

∑

0

⁢

<

d

∣

⁢

n

d

=

∏

i

=

1

k

p

i

2

⁢

(

m

i

+

1

)

-

1

p

i

2

-

1

:

∏

i

=

1

k

p

i

m

i

+

1

-

1

p

i

-

1

is an integer; here the

p

i

’s are the distinct

prime divisors

of

n

and

m

i

’s their

multiplicities

.  The last form is
simplified to

∏

i

=

1

k

p

i

m

i

+

1

+

1

p

i

+

1

.

(1)

The OEIS

sequence

A020487 contains all nonzero

perfect squares

,
since in the case of such numbers the antiharmonic mean (1) of
the divisors has the form

∏

i

=

1

k

p

i

2

⁢

m

i

+

1

+

1

p

i

+

1

=

∏

i

=

1

k

(

p

i

2

⁢

m

i

-

p

i

2

⁢

m

i

-

1

-

+

…

-

p

i

+

1

)

(cf.

irreducibility of binomials with unity coefficients

).

Note.

It would in a manner be legitimated to define
a positive integer to be an antiharmonic number (or an
antiharmonic integer) if it is the antiharmonic mean of two
distinct positive integers; see

integer contraharmonic mean

and

contraharmonic Diophantine equation

(

http://planetmath.org/ContraharmonicDiophantineEquation

).

Title

antiharmonic number

Canonical name

AntiharmonicNumber

Date of creation

2013-11-28 10:15:29

Last modified on

2013-11-28 10:15:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 11A05

Classification

msc 11A25

Generated on Fri Feb 9 15:09:19 2018 by

LaTeXML
Bezout’s lemma (number theory)

Bezout’s lemma (number theory)

Let

a

,

b

be

integers

, not both zero. Then there exist two integers

x

,

y

such that:

a

⁢

x

+

b

⁢

y

=

gcd

⁡

(

a

,

b

)

.

This does not only work on

ℤ

but on every

integral domain

where an

Euclidean valuation

has been defined.

Title

Bezout’s lemma (

number theory

)

Canonical

name

BezoutsLemmanumberTheory

Date of creation

2013-03-22 12:40:40

Last modified on

2013-03-22 12:40:40

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

10

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 11A05

Synonym

Bezout’s lemma

Synonym

Bezout’s theorem

Related topic

EuclidsAlgorithm

Related topic

EuclidsCoefficients

Related topic

GreatestCommonDivisorOfSeveralIntegers

Generated on Fri Feb 9 15:09:21 2018 by

LaTeXML
colossally abundant number

colossally abundant number

An

integer

n

is a

colossally abundant number

if there is an exponent

ϵ

>

1

such that the sum of

divisors

of

n

divided by

n

raised to that exponent is greater than or equal to the sum of divisors of any other integer

k

>

1

divided by

k

raised to that same exponent. That is,

σ

⁢

(

n

)

n

ϵ

≥

σ

⁢

(

k

)

k

ϵ

,

with

σ

⁢

(

n

)

being the

sum of divisors function

.

The first few colossally abundant numbers are 1, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320. The index of a colossally abundant number is equal to the number of its nondistinct

prime factors

, that is to say that for the

i

th colossally abundant number

c

i

the equality

i

=

Ω

⁢

(

c

i

)

is true.

Title

colossally abundant number

Canonical name

ColossallyAbundantNumber

Date of creation

2013-03-22 17:37:52

Last modified on

2013-03-22 17:37:52

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A05

Generated on Fri Feb 9 15:09:22 2018 by

LaTeXML
congruence of arbitrary degree

congruence of arbitrary degree

Theorem

.

A

congruence

of

n

th degree and modulo a

prime number

has at most

n

incongruent roots.

Proof.

In the case

n

=

1

,  the assertion turns out from the entry

linear congruence

.  We make the

induction hypothesis

, that the assertion is true for congruences of degree less than

n

.

We suppose now that the congruence

f

⁢

(

x

)

:=

a

n

⁢

x

n

+

a

n

-

1

⁢

x

n

-

1

+

…

+

a

0

≡

0

(

mod

p

)

,

(1)

where

p

∤

a

n

,  has at least

n

incongruent roots

x

1

,

x

2

,

…

,

x

n

.  Form the congruence

f

⁢

(

x

)

≡

a

n

⁢

(

x

-

x

1

)

⁢

(

x

-

x

2

)

⁢

⋯

⁢

(

x

-

x

n

)

(

mod

p

)

.

(2)

Both sides have the same term

a

n

⁢

x

n

of the highest degree, whence they may be cancelled from the congruence and the degree of (2) has a lower degree than

n

.  Because (2), however, clearly has

n

incongruent roots

x

1

,

x

2

,

…

,

x

n

,  it must by the induction hypothesis be simplifiable to the form

0

≡

0

(

mod

p

)

and thus be an

identical congruence

.

Now, if the congruence (1) had an additional incongruent root

x

n

+

1

, i.e.

P

⁢

(

x

n

+

1

)

≡

0

(

mod

p

)

, then the identical congruence (2) would imply

a

n

⁢

(

x

n

+

1

-

x

1

)

⁢

(

x

n

+

1

-

x

2

)

⁢

⋯

⁢

(

x

n

+

1

-

x

n

)

≡

0

(

mod

p

)

.

Yet, this is impossible, since no one of the factors (

http://planetmath.org/

Product

) of the left hand side is

divisible

by

p

.  This settles the

induction

proof.

Cf.

http://eom.springer.de/c/c024860.htm

SpringerLink.

Example.

When

f

⁢

(

x

)

:=

x

5

+

x

+

1

≡

0

(

mod

7

)

,  we have

f

⁢

(

0

)

≡

1

(

mod

7

)

,

f

⁢

(

1

)

≡

3

(

mod

7

)

,

f

⁢

(

2

)

≡

32

+

2

+

1

≡

0

(

mod

7

)

,

f

⁢

(

3

)

≡

27

⋅

9

+

3

+

1

≡

-

1

⋅

2

+

4

≡

2

(

mod

7

)

,

f

⁢

(

4

)

≡

(

-

3

)

5

+

4

+

1

≡

+

2

+

5

≡

0

(

mod

7

)

,

f

⁢

(

5

)

≡

(

-

2

)

5

+

5

+

1

≡

-

32

+

6

≡

-

26

≡

2

(

mod

7

)

,

f

⁢

(

6

)

≡

(

-

1

)

5

+

6

+

1

≡

6

(

mod

7

)

.

Thus only the representants 2 and 4 of a

complete residue system

modulo 7 (see

conditional congruences

) are roots of the given congruense.  A congruence needs not have the maximal amount of incongruent roots mentionned in the theorem.

References

1

K. Väisälä

:

Lukuteorian ja korkeamman algebran alkeet

.  Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).

Title

congruence of arbitrary degree

Canonical

name

CongruenceOfArbitraryDegree

Date of creation

2013-03-22 18:52:29

Last modified on

2013-03-22 18:52:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A05

Classification

msc 11A07

Related topic

SufficientConditionOfPolynomialCongruence

Related topic

APolynomialOfDegreeNOverAFieldHasAtMostNRoots

Generated on Fri Feb 9 15:09:25 2018 by

LaTeXML
corollary of Bézout’s lemma

corollary of Bézout’s lemma

Theorem

.

If

gcd

⁡

(

a

,

c

)

=

1

and

c

|

a

b

,  then

c

|

b

.

Proof.

Bézout’s lemma (

http://planetmath.org/BezoutsLemma

) gives the

integers

x

and

y

such that

x

⁢

a

+

y

⁢

c

=

1

.  This implies that

x

⁢

a

⁢

b

+

y

⁢

b

⁢

c

=

b

,  and because here the both summands are

divisible

by

c

, so also the sum, i.e.

b

, is divisible by

c

.

Note.

A similar theorem holds in all Bézout domains (

http://planetmath.org/BezoutDomain

), also in

Bézout rings

.

Title

corollary of Bézout’s lemma

Canonical name

CorollaryOfBezoutsLemma

Date of creation

2013-03-22 14:48:16

Last modified on

2013-03-22 14:48:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A05

Synonym

Euclid’s lemma

Synonym

product divisible but factor coprime

Related topic

GreatestCommonDivisor

Related topic

DivisibilityInRings

Related topic

DivisibilityByProduct

Generated on Fri Feb 9 15:09:27 2018 by

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deficiency

deficiency

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

2

⁢

n

-

(

∑

i

=

1

k

d

i

)

is the

deficiency

of

n

. Or if one prefers,

n

-

(

∑

i

=

1

k

-

1

d

i

)

.

The deficiency is essentially the same thing as the

abundance

multiplied by

-

1

. Thus, the deficiency is

positive

for

deficient numbers

, 0 for

perfect numbers

and negative for

abundant numbers

.

For example, the divisors of 13 add up to 14, which is 12 less than 26. Therefore, 12 has an deficiency of 12. Another example: the divisors of 14 add up to 24, which is 4 less than 28. The deficiency of the first 72 integers is listed in A033879 of Sloane’s OEIS.

Title

deficiency

Canonical

name

Deficiency

Date of creation

2013-03-22 16:46:50

Last modified on

2013-03-22 16:46:50

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A05

Related topic

Abundance

Generated on Fri Feb 9 15:09:28 2018 by

LaTeXML
deficient number

deficient number

A

deficient number

is an

integer

n

such that its

proper divisors

add up to less than itself, or all its

divisors

add up to less than twice itself. For example, 26. Its proper divisors are 1, 2 and 13, which add up to 16, which is 10 short of 26. Or if we also add 26, the divisors add up to 42, which is 10 short of 52.

All

prime numbers

are deficient, since 1 is their only proper divisor. With

σ

⁢

(

n

)

being the

sum of divisors function

, we can write that for a prime number

p

it is always the case that

σ

⁢

(

p

)

=

p

+

1

. Thanks to Euclid’s proof of the infinitude of primes, it is also proven that there are infinitely many deficient numbers.

An integer

power of two

(

2

x

for

x

>

0

) is always deficient, since its proper divisors add up to

2

x

-

1

.

Given a pair of

amicable numbers

, the greater of the two is deficient and its proper divisors add up to the smaller of the two, while the lesser of the two is an

abundant number

with its proper divisors adding up to the larger of the two.

Title

deficient number

Canonical name

DeficientNumber

Date of creation

2013-03-22 15:52:18

Last modified on

2013-03-22 15:52:18

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A05

Synonym

defective number

Related topic

AmicableNumbers

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

LaTeXML
determination of abundant numbers with specified prime factors

determination of abundant numbers with specified prime factors

The

formula

for sums of factors may be used to
find all

abundant numbers

with a specified set
of

prime factors

or that no such numbers
exist. To accomplish this, we first do a
little

algebraic

manipulation to our formula.

Theorem 1

.

A number

n

whose factorization
into

prime numbers

is

∏

i

=

1

k

p

i

m

1

is abundant if and only if

∏

i

=

1

k

(

1

-

p

i

-

m

i

-

1

1

-

p

i

-

1

)

>

2

.

Proof.

By definition

n

is abundant, if the sum of
the

proper divisors

of

n

is

greater

than

n

.
Using our formula, this is

equivalent

to the condition

∏

i

=

1

k

(

p

i

m

i

+

1

-

1

p

i

-

1

)

>

2

⁢

∏

i

=

1

k

p

i

m

i

.

Dividing the

k

-th term in the

product

on
the left-hand side by the

k

-th term on the
right-hand side,

1

p

i

m

i

⁢

p

i

m

i

+

1

-

1

p

i

-

1

=

p

i

-

m

i

-

1

p

i

-

1

⁢

p

i

-

p

i

-

m

i

p

i

-

1

=

1

-

p

i

-

m

i

-

1

1

-

p

i

-

1

,

so the condition becomes

∏

i

=

1

k

(

1

-

p

i

-

m

i

-

1

1

-

p

i

-

1

)

>

2

∎

Note that each of the terms in the product is bigger than 1.
Furthemore, the

k

-th term is

bounded

by

1

1

-

p

i

-

1

=

p

i

p

i

-

1

.

This means that it is only possible to have an abundant
number whose prime factors are

{

p

i

∣

1

≤

i

≤

k

}

if

∏

i

=

1

k

(

p

i

p

i

-

1

)

>

2

.

As it turns out, the convers also holds, so we have a
nice criterion for determining when a set of prime
numbers happens to be the set of

prime divisors

of
an abundant number.

Theorem 2

.

A

finite set

S

of prime numbers is the set of
prime divisors of an abundant number if and only if

∏

p

∈

S

(

p

p

-

1

)

>

2

.

Proof.

As described above, if

S

is a set of prime factors
of an abundant number, then we may bound each term
in the

inequality

of the previous

theorem

to obtain
the inequality in the current theorem. Assume, then
that

S

is a finite set of prime numbers which

satisfies

said inequality. Then, by

continuity

,
there must exist a

real number

ϵ

>

0

such
that

∏

p

∈

S

(

p

p

-

1

-

x

)

>

2

whenver

0

<

x

<

ϵ

. Since

lim

k

→

∞

⁡

n

-

k

=

0

when

n

>

1

, we can, for every

p

∈

S

,
find an

m

⁢

(

p

)

such that

p

m

⁢

(

p

)

p

-

1

<

ϵ

.

Hence,

∏

p

∈

S

(

p

p

-

1

-

p

m

⁢

(

p

)

p

-

1

)

=

∏

p

∈

S

(

1

-

p

-

m

⁢

(

p

)

-

1

1

-

p

-

1

)

>

2

so, by the previous theorem,

∏

p

∈

S

p

m

⁢

(

p

)

is an abundant number.
∎

Title

determination of abundant numbers with specified prime factors

Canonical name

DeterminationOfAbundantNumbersWithSpecifiedPrimeFactors

Date of creation

2013-03-22 16:47:41

Last modified on

2013-03-22 16:47:41

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 11A05

Generated on Fri Feb 9 15:09:32 2018 by

LaTeXML
divisibility by prime number

divisibility by prime number

Theorem

.

Let

a

and

b

be

integers

and

p

any

prime number

.  Then we have:

p

∣

a

b

⇔

p

∣

a

∨

p

∣

b

(1)

Proof.

Suppose that

p

∣

a

b

.  Then either

p

∣

a

or

p

∤

a

.  In the latter case we have

gcd

⁡

(

a

,

p

)

=

1

,  and therefore the corollary of Bézout’s lemma gives the result

p

∣

b

.  Conversely, if

p

∣

a

or

p

∣

b

,  then for example

a

=

m

⁢

p

for some integer

m

; this implies that

a

⁢

b

=

m

⁢

b

⋅

p

,  i.e.

p

∣

a

b

.

Remark 1.

The theorem means, that if a product is

divisible

by a prime number, then at least one of the factor is divisibe by the prime number. Also conversely.

Remark 2.

The condition (1) is expressed in of

principal ideals

as

(

a

⁢

b

)

⊆

(

p

)

⇔

(

a

)

⊆

(

p

)

∨

(

b

)

⊆

(

p

)

.

(2)

Here,

(

p

)

is a

prime ideal

of

ℤ

.

Title

divisibility by prime number

Canonical name

DivisibilityByPrimeNumber

Date of creation

2013-03-22 14:48:18

Last modified on

2013-03-22 14:48:18

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A05

Synonym

divisibility by prime

Related topic

PrimeElement

Related topic

DivisibilityInRings

Related topic

EulerPhiAtAProduct

Related topic

RepresentantsOfQuadraticResidues

Generated on Fri Feb 9 15:09:39 2018 by

LaTeXML
divisor function

divisor function

In the parent article there has been proved the

formula

σ

1

⁢

(

n

)

=

∑

0

⁢

<

d

∣

⁢

n

d

=

∏

i

=

1

k

p

i

m

i

+

1

-

1

p

i

-

1

giving the sum of all

positive

divisors

of an

integer

n

;
there the

p

i

’s are the distinct positive

prime factors

of

n

and

m

i

’s their

multiplicities

.

It follows that the sum of the

z

’th powers of those divisors is given by

σ

z

⁢

(

n

)

=

∑

0

⁢

<

d

∣

⁢

n

d

z

=

∏

i

=

1

k

p

i

(

m

i

+

1

)

⁢

z

-

1

p

i

z

-

1

.

(1)

This complex function of

z

is called

divisor function

(

http://planetmath.org/DivisorFunction

)

.  The
equation (1) may be written in the form

σ

z

⁢

(

n

)

=

∏

i

=

1

k

(

1

+

p

i

z

+

p

i

2

⁢

z

+

…

+

p

i

m

i

⁢

z

)

(2)

usable also for

z

=

0

.  For the special case of one prime
power the

function

consists of the single
geometric sum (

http://planetmath.org/GeometricSeries

)

σ

z

⁢

(

p

m

)

=

1

+

p

z

+

p

2

⁢

z

+

…

+

p

m

⁢

z

,

which particularly gives

m

+

1

when

p

z

=

1

, i.e. when

z

is a

multiple

of

2

⁢

i

⁢

π

/

ln

⁡

p

.

A special case of the function (1) is the

τ

function (

http://planetmath.org/TauFunction

) of

n

:

σ

0

⁢

(

n

)

=

∑

0

⁢

<

d

∣

⁢

n

1

=

∏

i

=

1

k

(

m

i

+

1

)

=

τ

⁢

(

n

)

Some inequalities

σ

m

⁢

(

n

)

≥

n

m

2

⁢

σ

0

⁢

(

n

)

for

m

=

0

,

1

,

2

,

…

σ

1

⁢

(

m

⁢

n

)

>

σ

1

⁢

(

m

)

+

σ

1

⁢

(

n

)

∀

m

,

n

∈

ℤ

σ

1

⁢

(

n

)

≤

n

+

1

2

⁢

σ

0

⁢

(

n

)

n

+

n

<

σ

1

⁢

(

n

)

<

6

π

2

⋅

n

⁢

n

Title

divisor function

Canonical name

DivisorFunction

Date of creation

2013-11-27 18:13:38

Last modified on

2013-11-27 18:13:38

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A05

Classification

msc 11A25

Generated on Fri Feb 9 15:09:41 2018 by

LaTeXML
Erdős-Woods number

Erdős-Woods number

An

integer

k

is an

Erdős-Woods number

if there is an integer

n

such that each of the consecutive integers

n

+

i

for

0

<

i

<

k

shares at least one

prime factor

with either

n

or

n

+

k

. In other words, if for a

k

there is an

n

such that each evaluation of

gcd

⁡

(

n

,

n

+

i

)

>

1

or

gcd

⁡

(

n

+

k

,

n

+

i

)

>

1

returns true, then

k

is an Erdős-Woods number.

For example, one

n

for

k

=

16

is 2184. 2184 is

2

3

×

3

×

7

×

13

, while

2184

+

16

=

2200

=

2

3

×

5

2

×

11

. We then verify that

•

2185 is clearly

divisible

by 5 and thus shares that

odd prime

as a factor with 2200.

•

2186 is even and so shares 2 as a factor with both 2184 and 2200.

•

2187 is 3 more than 2184 and therefore must also be divisible by 3. In fact, it is

3

7

.

•

2188 is even and so shares 2 as a factor with both 2184 and 2200, suggesting we needn’t look at any other

even numbers

in this range.

•

2189 is 11 less than 2200 and therefore must be divisible by 11. In base 10 we can quickly verify that 2 + 8 = 10 and 1 + 9 is also 10.

•

2191 is 7 more than 2184 and thus must be divisible by 7.

•

2193 is 9 more than 2184 and thus divisible by 3.

•

2195 is obviously divisible by 5.

•

2197 is 13 more than 2184 and thus must be divisible by 13. In fact, it is

13

3

.

•

2199 is 15 more than 2184 and thus divisible by 3.

Knowing one

n

for a given

k

one can find other

n

by multiplying the odd prime factors of

n

+

k

(just once each, let’s call that product “

q

”) and then calculating

n

⁢

(

j

⁢

q

+

1

)

with

j

any

positive integer

of one’s choice. To give one example: with

n

=

2184

and

j

=

17

, we get another

n

, namely 2044224. The range 2044224 to 2044240 displays the same patterns of factorization as described above, except that 2044227 and 2044237 are a

semiprime

and a

sphenic number

respectively, as opposed to 2187 and 2197 which are both prime powers.

Other Erdős-Woods numbers are 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92, 94, 96, 100, etc., listed in A059756 of Sloane’s OEIS (the smallest odd Erdős-Woods number is 903), while A059757 lists the smallest

matching

n

for each of those

k

.

References

1

R. K. Guy,

Unsolved Problems in

Number Theory

New York: Springer-Verlag 2004: B28

Title

Erdős-Woods number

Canonical

name

ErdHosWoodsNumber

Date of creation

2013-03-22 17:37:14

Last modified on

2013-03-22 17:37:14

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A05

Synonym

Erdos-Woods number

Synonym

Erdös-Woods number

Generated on Fri Feb 9 15:09:43 2018 by

LaTeXML
Euclid’s coefficients

Euclid’s coefficients

The following program is based on Euclid’s algorithm and it determines

Euclid’s

coefficients

t

and

d

of the

integers

x

and

y

.

INPUT

x

,

y

(

positive integers

)

m

:=

x

,

n

:=

y

b

:=

0

,

d

:=

1

p

:=

1

,

t

:=

0

WHILE

m

≠

0

DO

BEGIN

q

:=

n

DIV

m

(integer division)

h

:=

m

,

m

:=

n

-

q

⁢

m

,

n

:=

h

h

:=

b

,

b

:=

d

-

q

⁢

b

,

d

:=

h

h

:=

p

,

p

:=

t

-

q

⁢

p

,

t

:=

h

END

WRITE   — The gcd of the numbers

x

and

y

is

n

=

t

⁢

x

+

d

⁢

y

.

Remark.

The values of

t

and

d

produced by the program have the

absolute values

as close each other as possible (Proof = ?).  They differ very often only by 1 when

x

and

y

are two successive

prime numbers

, e.g.

1

=

30

⋅

523

-

29

⋅

541

.

Title

Euclid’s coefficients

Canonical name

EuclidsCoefficients

Date of creation

2013-03-22 15:15:20

Last modified on

2013-03-22 15:15:20

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Algorithm

Classification

msc 11A05

Classification

msc 03-04

Related topic

BezoutsLemma

Generated on Fri Feb 9 15:09:48 2018 by

LaTeXML