aarjaneiro/planetmath huggingface dataset:
Achilles number

Achilles number

Given a

composite number

n

with factorization

∏

i

=

1

ω

⁢

(

n

)

p

i

a

i

,

where the

p

i

are all distinct primes, and the

a

i

are

positive integers

(and also distinct, always

a

i

>

1

), and

ω

⁢

(

x

)

is the

number of distinct prime factors function

, then

n

is a

squarefull number

, and if it is also the case that there is no solution to

n

=

x

y

in

integers

(with

x

≠

n

and

y

>

1

), then

n

is called an

Achilles number

.

So, for example, 1323, being

3

3

×

7

2

, is

divisible

by the squares of both 3 and 7, but its

square root

is approximately 36.373, its cubic root is about 10.9779, its fourth root is about 6.031, its fifth root is about 4.21, its sixth root is 3.313, seventh root about 2.792, etc.

An Achilles number has to have distinct

exponents

for its

prime factors

: with equal exponents, we can see that

p

a

x

⁢

p

b

x

=

(

p

a

⁢

p

b

)

x

. The reverse is often not true: for example, 144 is

2

4

×

3

2

, yet it can be expressed as

12

2

.

According to Greek legend, Achilles was a powerful, seemingly invincible warrior who fought in the Trojan War. In some versions of the story, his mother Thetis held baby Achilles by the left heel and dipped him into the Styx river, making him invulnerable to attack anywhere on his body except the one heel. Thus, Achilles was very powerful, but not perfect because of his fatal weak spot. Likewise, Achilles numbers are powerfull but they are not

perfect powers

.

References

1

C. Bossley, B. Martinelli, N. Maffulli, & C. Raisbeck, “Rupture of the Achilles Tendon”

J Bone Joint Surg Am.

2000 Dec; 82 - A (12): 1804

2

H. Bottomley & N. Sloane,

Sequence

A052486 in

On-Line Encyclopedia of Integer Sequences

Title

Achilles number

Canonical

name

AchillesNumber

Date of creation

2013-03-22 17:10:33

Last modified on

2013-03-22 17:10:33

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Generated on Fri Feb 9 15:19:13 2018 by

LaTeXML
beprisque number

beprisque number

A

beprisque number

n

is an

integer

which is either one more than a

prime number

and one less than a

perfect square

, or one more than a square and one less than a prime. That is, either

(

n

-

1

)

∈

ℙ

and

n

+

1

∈

ℤ

or

n

-

1

∈

ℤ

and

(

n

+

1

)

∈

ℙ

.

The beprisque numbers below a thousand are 1, 2, 3, 8, 10, 24, 48, 80, 82, 168, 224, 226, 360, 440, 442, 728, 840. The listing on the OEIS, A163492, goes further but omits 1. It could be argued that a beprisque number should itself not be a square, but then it becomes difficult to justify why beprisque numbers that are themselves primes are allowed.

The only prime beprisque numbers are 2 and 3. For a prime number to be a beprisque number, it has to neighbor another prime number; the only two primes fitting that bill are of course 2 and 3.

The only odd beprisque numbers are 1 and 3. For an

odd number

to be a beprisque number, it has to neighbor an

even prime

, and since 2 is the only even prime, only 1 and 3 can be beprisque numbers.

Title

beprisque number

Canonical name

BeprisqueNumber

Date of creation

2013-03-22 19:10:11

Last modified on

2013-03-22 19:10:11

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Generated on Fri Feb 9 15:19:14 2018 by

LaTeXML
Blum number

Blum number

Given a

semiprime

n

=

p

⁢

q

, if both

p

and

q

are

Gaussian primes

with no

imaginary part

, then

n

is called a

Blum number

. The first few Blum numbers are 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, etc., listed in A016105 of Sloane’s OEIS.

A semiprime that is a Blum number is also a semiprime among the

Gaussian integers

and its

prime factors

also have no imaginary parts. The other real semiprimes are not semiprimes among the Gaussian integers. For example, 177 can only be factored as

3

×

59

whether Gaussian integers are allowed or not. 159, on the other hand can be factored as either

3

×

53

or

3

⁢

(

-

i

)

⁢

(

2

+

7

⁢

i

)

⁢

(

7

+

2

⁢

i

)

.

Large Blum numbers had applications in cryptography prior to advances in

integer factorization

by means of

quadratic sieves

.

Title

Blum number

Canonical name

BlumNumber

Date of creation

2013-03-22 17:53:17

Last modified on

2013-03-22 17:53:17

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Synonym

Blum integer

Generated on Fri Feb 9 15:19:16 2018 by

LaTeXML
Chinese hypothesis

Chinese hypothesis

The

Chinese hypothesis

states that a number

n

is prime if and only if

2

n

-

2

is a

multiple

of

n

. By Fermat’s little

theorem

we have that

2

p

≡

2

mod

p

, so that means

n

does divide

2

n

-

2

if

n

is prime. However, if

n

is

composite

Fermat’s little theorem does not rule out that

n

could divide

2

n

-

2

. The Chinese hypothesis checks out for the small

powers of two

. The first

counterexample

is

n

=

341

, but since

2

341

has more than a

hundred

digits, it wasn’t easy to check it back in the 18th century when this test was first proposed. Though back then they attributed it to ancient Chinese mathematicians, hence the name.

Title

Chinese hypothesis

Canonical

name

ChineseHypothesis

Date of creation

2013-03-22 18:11:08

Last modified on

2013-03-22 18:11:08

Owner

FourDozens (21006)

Last modified by

FourDozens (21006)

Numerical id

6

Author

FourDozens (21006)

Entry type

Definition

Classification

msc 11A51

Generated on Fri Feb 9 15:19:17 2018 by

LaTeXML
Chinese remainder theorem in terms of divisor theory

Chinese remainder theorem in terms of divisor theory

In a ring with a

divisor theory

, a

congruence

α

≡

β

(

mod

𝔞

)

with respect to a

divisor

module (

http://planetmath.org/Congruences

)

𝔞

that

𝔞

∣

α

-

β

.

Theorem.

Let

𝒪

be an

integral domain

having the divisor theory

𝒪

*

→

𝔇

.  For arbitrary

pairwise coprime

divisors

𝔞

1

,

…

,

𝔞

s

in

𝔇

and for arbitrary elements

α

1

,

…

,

α

s

of the domain

𝒪

there exists an element

ξ

in

𝒪

such that

{

ξ

≡

α

1

(

mod

𝔞

1

)

⋯

⋯

⋯

ξ

≡

α

s

(

mod

𝔞

s

)

Proof.

Let

𝔟

i

:=

∏

j

≠

i

𝔞

j

(

i

=

1

,

…

,

s

)

.

Apparently, the divisors

𝔟

1

,

…

,

𝔟

s

are mutually coprime, whence there are in the ring

𝒪

the elements

β

1

,

…

,

β

s

divisible

by  the divisors

𝔟

1

,

…

,

𝔟

s

,  respectively, such that

β

1

+

…

+

β

s

=

1

.

(1)

For every

i

≠

j

,  the divisor

𝔞

i

divides

𝔟

j

and therefore also the element

β

j

.  Then the equation (1) implies that

β

i

≡

1

(

mod

𝔞

i

)

and thus the element

ξ

:=

α

1

⁢

β

1

+

…

+

α

s

⁢

β

s

satisfies

ξ

≡

α

i

⁢

β

i

≡

α

i

(

mod

𝔞

i

)

for each

i

=

1

,

…

,

s

.  Q.E.D.

References

1

М. М. Постников:

Введение  в  теорию  алгебраических
чисел

.  Издательство
‘‘Наука’’. Москва (1982).

Title

Chinese remainder theorem in terms of divisor theory

Canonical name

ChineseRemainderTheoremInTermsOfDivisorTheory

Date of creation

2013-03-22 18:01:58

Last modified on

2013-03-22 18:01:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A51

Classification

msc 13A05

Related topic

ChineseRemainderTheorem

Related topic

ChineseRemainderTheorem2

Related topic

CongruenceInAlgebraicNumberField

Related topic

WeakApproximationTheorem

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

LaTeXML
Cullen number

Cullen number

A

Cullen number

C

m

is a number of the form

2

m

⁢

m

+

1

, where

m

is an

integer

. Almost all Cullen numbers are

composite

, and they have

2

⁢

m

-

1

as a factor if that number is a prime of the form

8

⁢

k

-

3

(with

k

a

positive integer

). The first few Cullen numbers are 1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609 (listed in A002064 of Sloane’s OEIS). With the exception of 3, no

Proth number

is also a Cullen number.

Title

Cullen number

Canonical name

CullenNumber

Date of creation

2013-03-22 17:21:34

Last modified on

2013-03-22 17:21:34

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Related topic

WoodallNumber

Generated on Fri Feb 9 15:19:25 2018 by

LaTeXML
Cullen prime

Cullen prime

A

Cullen prime

is a

Cullen number

C

m

that is also a

prime number

. The first two Cullen primes are 3 and 393050634124102232869567034555427371542904833, corresponding to

m

=

1

and 141 respectively; other

m

to give Cullen primes are 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, etc. listed in A005849 of Sloane’s OEIS (those primes being too large to write here). None of these indexes are prime, and it is a matter of

conjecture

whether there exists a Cullen prime of the form

p

⁢

2

p

+

1

with

p

also prime. It is not even known whether there are infinitely many Cullen primes (a possibility despite their rarity).

Title

Cullen prime

Canonical

name

CullenPrime

Date of creation

2013-03-22 17:21:37

Last modified on

2013-03-22 17:21:37

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Generated on Fri Feb 9 15:19:27 2018 by

LaTeXML
de Polignac’s formula

de Polignac’s formula

Given

n

, the

prime factorization

of

n

!

can be obtained by applying

de Polignac’s formula

:

∏

i

=

1

π

⁢

(

n

)

p

i

∑

j

=

1

⌊

log

p

i

⁡

n

⌋

⌊

n

p

i

j

⌋

,

where

p

i

is the

i

th prime and

π

⁢

(

n

)

is the

prime counting function

. For both the product and the

summation

, the iterator’s end value can be set to infinity and the formula is still correct. When

n

<

p

i

j

, dividing the former by the latter gives a value that floors to zero, and then

p

i

0

=

1

. And when

i

>

π

⁢

(

n

)

then all

j

give zero in the summation, and the multiplicand for the product is also 1. It is for the sake of a

computer

implementation that it is necessary to stop the iterations when they no longer give useful results.

A small disadvantage of de Polignac’s formula is that it is necessary to know all the primes up to

n

. If the implementation mistakenly iterates through a

composite

value, the error might not be detected, while even with a dumb implementation of

trial division

(e.g., one that tries odd composite values) the worst that can happen is that time is wasted by trying to divide by a composite value whose

prime factors

have already been divided out of the

factorial

.

Let’s work out an example:

factoring

10!. The primes less than 10 are 2, 3, 5 and 7. The base 2 logarithm of 10 is approximately 3.32, meaning we only need to divide and floor up to 2’s cube. Half 10 is 5, a quarter of 10 is 2.5 and an eighth of 10 is 1.25; adding up the

integer parts

of these gives us 8, so the exponent for 2 in the factorization of 10! is 8. The base 3 logarithm of 10 is approximately 2.0959. A third of 10 is about 3.3333 and a ninth of 10 is about 1.1111; the integer parts of these added up gives 4, so the exponent for 3 in the factorization is 4. For both bases 5 and 7, the logarithm of 10 is more than 1 but less than 2. 10 divided by 5 is 2, so that’s our exponent for 5. 10 divided by 7 is about 1.42857, so 1 is our exponent for 7. Then we verify that indeed

2

8

×

3

4

×

5

2

×

7

1

=

3628800

=

10

!

Title

de Polignac’s formula

Canonical

name

DePolignacsFormula

Date of creation

2013-03-22 17:59:54

Last modified on

2013-03-22 17:59:54

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Generated on Fri Feb 9 15:19:29 2018 by

LaTeXML
divisibility

divisibility

Given

integers

a

and

b

, then we say

a

divides

b

if and only if there is some

q

∈

ℤ

such that

b

=

q

⁢

a

.

There are many other ways in common use to express this relationship:

•

a

∣

b

(read “

a

divides

b

”).

•

b

is

divisible

by

a

.

•

a

is a factor of

b

.

•

a

is a

divisor

of

b

.

•

b

is a

multiple

of

a

.

The notion of divisibility can apply to other rings (e.g.,

polynomials

).

Title

divisibility

Canonical name

Divisibility

Date of creation

2013-03-22 11:59:49

Last modified on

2013-03-22 11:59:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11A51

Synonym

divides

Synonym

divisor

Synonym

factor

Synonym

multiple

Related topic

LeastCommonMultiple

Related topic

ExampleOfGcd

Related topic

TauFunction

Related topic

ExactlyDivides

Related topic

DivisorSumOfAnArithmeticFunction

Related topic

StrictDivisibility

Related topic

FundamentalTheoremOfArithmetic

Related topic

NumberTheory

Generated on Fri Feb 9 15:19:30 2018 by

LaTeXML
divisibility by product

divisibility by product

Theorem

.

Let

R

be a

Bézout ring

, i.e. a

commutative ring

with

non-zero unity

where every

finitely generated

ideal is a

principal ideal

. If

a

,

b

,

c

are three elements of

R

such that

a

and

b

divide

c

and

gcd

⁡

(

a

,

b

)

=

1

,  then also

a

⁢

b

divides

c

.

Proof.

The

divisibility

assumptions that

c

=

a

⁢

a

1

=

b

⁢

b

1

where

a

1

and

b

1

are some elements of

R

.  Because

R

is a Bézout ring, there exist such elements

x

and

y

of

R

that

gcd

⁡

(

a

,

b

)

=

1

=

x

⁢

a

+

y

⁢

b

. This implies the

equation

a

1

=

x

⁢

a

⁢

a

1

+

y

⁢

b

⁢

a

1

=

x

⁢

b

⁢

b

1

+

y

⁢

b

⁢

a

1

which shows that

a

1

is

divisible

by

b

, i.e.

a

1

=

b

⁢

b

2

,

b

2

∈

R

. Consequently,

c

=

a

⁢

a

1

=

a

⁢

b

⁢

b

2

,  or

a

b

∣

c

Q.E.D.

Note 1.

The theorem may by induction be generalized for several factors (

http://planetmath.org/Divisibility

) of

c

.

Note 2.

The theorem holds e.g. in all Bézout domains, especially in

principal ideal domains

, such as

ℤ

and

polynomial rings

over a field.

Title

divisibility by product

Canonical name

DivisibilityByProduct

Date of creation

2013-03-22 14:50:37

Last modified on

2013-03-22 14:50:37

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A51

Classification

msc 13A05

Related topic

BezoutDomain

Related topic

ProductDivisibleButFactorCoprime

Related topic

CorollaryOfBezoutsLemma

Defines

Bézout ring

Generated on Fri Feb 9 15:19:32 2018 by

LaTeXML
division algorithm for integers

division algorithm for integers

Given any two

integers

a

,

b

where

b

>

0

, there exists a unique pair of integers

q

,

r

such that

a

=

q

⁢

b

+

r

and

0

≤

r

<

b

.

q

is called the

quotient

of

a

and

b

, and

r

is the

remainder

.

The

division algorithm

is not an algorithm at all but rather a theorem. Its name probably derives from the fact that it was first proved by showing that an algorithm to calculate the quotient of two integers yields this result.

There are similar forms of the division algorithm that apply to other rings (for example,

polynomials

).

Title

division algorithm for integers

Canonical name

DivisionAlgorithmForIntegers

Date of creation

2013-03-22 11:59:43

Last modified on

2013-03-22 11:59:43

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

7

Author

vampyr (22)

Entry type

Theorem

Classification

msc 11A51

Synonym

division algorithm

Related topic

ExistenceAndUniquenessOfTheGcdOfTwoIntegers

Generated on Fri Feb 9 15:19:34 2018 by

LaTeXML
Division Factorial

Division Factorial

Division

Factorial

1

1

Division Factorial

The prospect of the Division Factorial follows the same

concept

of

multiplicative

factor. The same

operations

known since the twelfth century. The same logical

properties

that would be studied later in 1808 by the mathematician Christian Kramp which introduced the notation n!. Stirling also presented a

formula

approach to these results.
However, it is urgent to talk about divisive factor. As you think of the beginning, not the multiplicative

inversion

extensive. It is

elementary

that the corporeality of Factor Theory arises simply. About

natural number

n which is the

product

of all

positive integers

less than or equal to n.
Demonstrate exemplary one truth and accuracy, which

supports

more than a demonstration

[

1

]

. Likewise, presents the multiplicative factor, which has

similar

properties thereto. Therefore, evidenced in the following series:

1

:

2

:

3

:

4

:

5

=

8333

⁢

…

×

10

3

applied by;

n

+

1

(

n

+

1

)

!

=

i

⁢

f

⁢

n

⁢

i

⁢

s

⁢

g

⁢

r

⁢

o

⁢

w

⁢

i

⁢

n

⁢

g

,

Otherwise, decreasing the following

sequence

:

5

:

4

:

3

:

2

:

1

=

208333

⁢

…

×

10

1

It is applied by:

n

(

n

-

1

)

!

=

i

⁢

f

⁢

n

⁢

i

⁢

s

⁢

d

⁢

e

⁢

c

⁢

r

⁢

e

⁢

a

⁢

s

⁢

i

⁢

n

⁢

g

,

References

1

FERNÃNDESKY, PAULO., 2013. Ös Teoremas. N. (Ed.). Escrytos of the

distribution

. Lisboa, 2013. p.22-38. (Statistics, Kindle, Artigo.

Title

Division Factorial

Canonical

name

DivisionFactorial

Date of creation

2013-10-30 22:15:55

Last modified on

2013-10-30 22:15:55

Owner

Paulo Fernandesky (1000738)

Last modified by

Paulo Fernandesky (1000738)

Numerical id

7

Author

Paulo Fernandesky (1000738)

Entry type

Theorem

Classification

msc 11A51 Division Factorial

Generated on Fri Feb 9 15:19:38 2018 by

LaTeXML
eliminated Sierpiński number candidates

eliminated Sierpiński number candidates

Most numbers

k

are very easy to eliminate as Sierpiński number candidates, as it is very easy to come up with

sequences

of primes of the form

k

⁢

2

n

+

1

. For example, for

k

=

1

it is enough to mention the

Fermat primes

. For some of the seventeen Sierpiński number candidates when the Seventeen or Bust project began, only a single

Proth prime

, and it is often quite large. Eight candidates remain to be eliminated. The primes listed below were discovered by the Seventeen or Bust project, with the latest being for 19249, discovered by user Konstantin Agafonov of team TSC! Russia.

k

Exponent

n

which gives prime

Prime in base 10

scientific notation

4847

3321063

1.844857508381060

×

10

999743

5359

5054502

2.781168752802502

×

10

1521560

10223

Still a candidate

N/A

19249

13018586

1.484360328715661

×

10

3918989

21181

Still a candidate

N/A

22699

Still a candidate

N/A

24737

Still a candidate

N/A

27653

9167433

5.727724120920733

×

10

2759676

28433

7830457

7.772839072447348

×

10

2357206

44131

Still a candidate

N/A

46157

Still a candidate

N/A

54767

Still a candidate

N/A

55459

995972

1.234767571821004

×

10

299822

65567

1013803

8.499227304459893

×

10

305189

67607

Still a candidate

N/A

69109

1157446

6.366429016367452

×

10

348430

Title

eliminated Sierpiński number candidates

Canonical name

EliminatedSierpinskiNumberCandidates

Date of creation

2013-03-22 17:21:14

Last modified on

2013-03-22 17:21:14

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A51

Generated on Fri Feb 9 15:19:46 2018 by

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Erdős-Selfridge classification of primes

Erdős-Selfridge classification of primes

Paul Erdős and John Selfridge classified primes

p

thus: If the largest prime factor of

p

+

1

is 2 or 3, then

p

is in class 1+. Otherwise, assign to

c

the class of the largest prime factor of

p

+

1

, then

p

is in class

(

c

+

1

)

+

. Class 1+ primes are of the form

2

i

⁢

3

j

-

1

for

i

>

-

1

and

j

>

-

1

.

According to this scheme,

p

<

200

are sorted thus:

Class 1+: 2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191 ( listed in A005105 of Sloane’s OEIS)

Class 2+: 13, 19, 29, 41, 43, 59, 61, 67, 79, 83, 89, 97, 101, 109, 131, 137, 139, 149, 167, 179, 197, 199 (A005106)

Class 3+: 37, 103, 113, 151, 157, 163, 173, 181, 193 (A005107)

Class 4+: 73

A005113 lists the smallest prime of class

n

+

.

Clearly, all

Mersenne primes

are class 1+. The known

Fermat primes

show slightly more variety: 257 is class 3+ while 65537 is class 4+.

References

1

R. K. Guy,

Unsolved Problems in

Number Theory

. New York: Springer-Verlag (2004)

Title

Erdős-Selfridge

classification

of primes

Canonical

name

ErdHosSelfridgeClassificationOfPrimes

Date of creation

2013-03-22 16:05:02

Last modified on

2013-03-22 16:05:02

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Synonym

Erdos-Selfridge classification of primes

Generated on Fri Feb 9 15:19:47 2018 by

LaTeXML
Euler pseudoprime

Euler pseudoprime

An

Euler pseudoprime

p

to a base

b

is a

composite number

for which the

congruence

b

p

-

1

2

≡

(

b

p

)

mod

p

holds true, where

(

a

n

)

is the

Jacobi symbol

.

For example, given

b

=

2

, our Jacobi symbol

(

2

p

)

with

p

odd will be either 1 or

-

1

. Then, for

p

=

561

, the Jacobi symbol is 1. Next, we see that 2 raised to the 280th is 1942668892225729070919461906823518906642406839052139521251812409738904285205208498176, which is one more than 561 times 3462867900580622229802962400754935662464183313818430519165441015577369492344400175. Hence 561 is an Euler pseudoprime. The next few Euler pseudoprimes to base 2 are 1105, 1729, 1905, 2047, 2465, 4033, 4681 (see A047713 in Sloane’s OEIS). An Euler pseudoprime is sometimes called an

Euler-Jacobi pseudoprime

, to distinguish it from a

pseudoprime

(

http://

planetmath

.org/PseudoprimeP

) for which the congruence can be either to 1 or

-

1

regardless of the Jacobi symbol (341 is then an Euler pseudoprime under this relaxed

definition

). Both terms are also sometimes used alone with 2 as the implied base.

If a composite number is an Euler pseudoprime to a given base, it is also a

regular

pseudoprime to that base, but not all regular pseudoprimes to that base are also Euler pseudoprimes to it.

References

1

R. Crandall & C. Pomerance,

Prime Numbers

: A Computational Perspective

, Springer, NY, 2001: 5.1

2

B. Fine & G. Rosenberger,

Number Theory

: An

Introduction

via the Distribution of the Primes

Boston: Birkhäuser, 2007: Definition 5.3.1.4

Title

Euler pseudoprime

Canonical

name

EulerPseudoprime

Date of creation

2013-03-22 16:49:48

Last modified on

2013-03-22 16:49:48

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Synonym

Euler-Jacobi pseudoprime

Generated on Fri Feb 9 15:19:49 2018 by

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exactly divides

exactly divides

Let

a

and

b

be

integers

and

n

a

positive integer

.  Then

a

m

exactly divides

b

(denoted as

a

n

∥

n

) if

a

n

divides

b

but

a

n

+

1

does not divide

b

.  For example,

2

4

∥

48

.

One can, of course, use the similar

expression

and notation for the elements

a

,

b

of any commutative ring or monoid (cf. e.g.

divisor as factor of principal divisor

).

Title

exactly divides

Canonical

name

ExactlyDivides

Date of creation

2013-03-22 16:10:44

Last modified on

2013-03-22 16:10:44

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11A51

Related topic

Divides

Related topic

Divisibility

Related topic

DivisibilityInRings

Generated on Fri Feb 9 15:19:51 2018 by

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