aarjaneiro/planetmath huggingface dataset:
proper divisor

proper divisor

If a

divisor

d

of

n

(that is,

d

|

n

) satisfies

0

<

|

d

|

<

|

n

|

, then

d

is a

proper divisor

of

n

. In the realm of real

positive integers

, it is usually considered sufficient to list the

positive

divisors. For example, the proper divisors of 42 are 1, 2, 3, 6, 7, 14, 21.

By restricting the sum of divisors to proper divisors, some

n

will be less than this sum (

deficient numbers

, including

prime numbers

), some will be equal (

perfect numbers

) and some will be greater (

abundant numbers

). The term

restricted divisor

is sometimes used to further distinguish divisors in the range

1

<

|

d

|

<

|

n

|

(and sometimes it used to mean the same thing as proper divisor). Thus, in our example, the list would be shortened to 2, 3, 6, 7, 14, 21.

Title

proper divisor

Canonical name

ProperDivisor

Date of creation

2013-03-22 15:52:00

Last modified on

2013-03-22 15:52:00

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Synonym

aliquot part

Synonym

restricted divisor

Generated on Fri Feb 9 15:20:39 2018 by

LaTeXML
Proth number

Proth number

A

Proth number

p

is a number of the form

k

⁢

2

m

+

1

, where

k

is odd (e.g.,

k

=

2

⁢

n

+

1

with an

integer

n

>

(

-

1

)

),

m

is also an integer and

2

m

>

k

. The first few Proth numbers are 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, etc. (listed in A080075 of Sloane’s OEIS). The

Fermat numbers

are a subset of the Proth numbers, namely with

k

=

1

.

Title

Proth number

Canonical name

ProthNumber

Date of creation

2013-03-22 17:21:09

Last modified on

2013-03-22 17:21:09

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Generated on Fri Feb 9 15:20:40 2018 by

LaTeXML
Proth prime

Proth prime

A

Proth prime

is a

Proth number

that is also a

prime number

. Given a Proth number

p

, if one can find a coprime integer

b

such that

b

p

-

1

2

≡

-

1

mod

p

then

p

is a prime, and specifically a Proth prime (this is a

theorem

first stated by François Proth). Thanks to this theorem, Yves Gallot created an algorithm used in a primality-testing program employed by the

Seventeen or Bust

project. The first few Proth primes are 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, etc. (listed in A080076 of Sloane’s OEIS). Konstantin Agafonov’s discovery of the Proth prime

19249

×

2

13018586

+

1

≈

1.484360328715661

×

10

3918989

currently makes for the largest known prime that is not a

Mersenne prime

.

Title

Proth prime

Canonical

name

ProthPrime

Date of creation

2013-03-22 17:21:11

Last modified on

2013-03-22 17:21:11

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Generated on Fri Feb 9 15:20:42 2018 by

LaTeXML
pseudoprime

pseudoprime

A

pseudoprime

is a

composite number

p

that passes a given

primality test

. In other words, a pseudoprime is a false

positive

on a primality test.

The primality test is usually a power

congruence

to a given base

b

that is

coprime

to

p

, such as

b

p

-

1

≡

1

mod

p

(from Fermat’s little theorem). For example, if

p

is an

odd prime

, then the congruence

2

p

-

1

≡

1

mod

p

will be true, but in reverse,

p

can satisfy the congruence but turn out to be an odd composite number. Such pseudoprimes are then called

pseudoprime to base

b

, and in our example with

b

=

2

, the first few are 341, 561, 645, 1105, 1387, 1729, 1905 (A001567 in Sloane’s OEIS). According to Neil Sloane, writing in the OEIS, the term “pseudoprime” is sometimes used specifically to refer to pseudoprimes to base 2. These are sometimes called

Sarrus numbers

.

Another kind of pseudoprimes are those in which a composite

p

divides an element it indexes in a Fibonacci-like

sequence

. For example, a

Perrin pseudoprime

p

divides

a

p

, where

a

n

is the

n

-th element in the

Perrin sequence

.

References

1

R. Crandall & C. Pomerance,

Prime Numbers

: A Computational Perspective

, Springer, NY, 2001: 5.1

Title

pseudoprime

Canonical name

Pseudoprime

Date of creation

2013-03-22 16:49:44

Last modified on

2013-03-22 16:49:44

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

6

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A51

Generated on Fri Feb 9 15:20:44 2018 by

LaTeXML
rational Briggsian logarithms of integers

rational Briggsian logarithms of integers

Theorem.

The only

positive integers

, whose

Briggsian logarithms

are

rational

, are the powers (

http://

planetmath

.org/GeneralAssociativity

)

1

,

10

,

100

,

…

of ten.  The

logarithms

of other positive integers are thus

irrational

(in fact,

transcendental numbers

).  The same concerns also the Briggsian logarithms of the

positive

fractional numbers

.

Proof.

Let

a

be a positive integer such that

lg

⁡

a

=

m

n

∈

ℚ

,

where

m

and

n

are positive integers.  By the definition of logarithm, we have

10

m

n

=

a

,  which is equivalent (

http://planetmath.org/Equivalent3

) to

10

m

=

2

m

⋅

5

m

=

a

n

.

According to the

fundamental theorem of arithmetics

, the

integer

a

n

must have exactly

m

prime divisors

2

and equally many prime divisors

5

.  This is not possible otherwise than that also

a

itself consists of a same amount of prime divisors 2 and 5, i.e. the number

a

is an integer power of 10.

As for any

rational number

a

b

(with

a

,

b

∈

ℤ

+

), if one had

lg

⁡

a

b

=

m

n

∈

ℚ

,

then

(

a

b

)

n

=

10

m

,

and it is apparent that the rational number

a

b

has to be an integer, more accurately a power of ten.  Therefore the logarithms of all fractional numbers are irrational.

Note.

An analogous theorem concerns e.g. the

binary logarithms

(

lb

⁡

a

).  As for the

natural logarithms

of positive rationals (

ln

⁡

a

), they all are transcendental numbers except

ln

⁡

1

=

0

.

Title

rational Briggsian logarithms of integers

Canonical name

RationalBriggsianLogarithmsOfIntegers

Date of creation

2013-03-22 17:39:55

Last modified on

2013-03-22 17:39:55

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A51

Related topic

Transcendental

Related topic

RationalSineAndCosine

Related topic

AllUnnaturalSquareRootsAreIrrational

Related topic

BriggsianLogarithms

Generated on Fri Feb 9 15:20:46 2018 by

LaTeXML
rough number

rough number

A

k

-

rough number

is an

integer

n

having only

prime factors

greater than or equal to

k

. A number that is

k

-rough is thus not a

k

-

smooth number

. However, a prime

p

is both

p

-smooth and

p

-rough because neither inequality is strict.

For example, 771166905830363 is 7-rough since, in

addition

to having 7 as a prime factor, it has greater primes as factors. By comparison, 93386641873154605056 is not 7-rough, and is in fact 7-smooth, since all its

divisors

are 2, 3, and powers of those small primes.

Title

rough number

Canonical

name

RoughNumber

Date of creation

2013-03-22 18:09:46

Last modified on

2013-03-22 18:09:46

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Related topic

SmoothNumber

Generated on Fri Feb 9 15:20:53 2018 by

LaTeXML
Ruth-Aaron pair

Ruth-Aaron pair

After Hank Aaron broke Babe Ruth’s home run record of 714, Carl Pomerance noticed that the

prime factors

of 714 and 715 both add up to 29. Since then, a pair of

integers

n

and

m

=

n

+

1

is called a

Ruth-Aaron pair

if the prime factors add up to the same number, and individually

n

is called a

Ruth-Aaron number

and so is

m

.

Given the factorizations of the numbers

n

=

∏

i

=

1

ω

⁢

(

n

)

p

i

a

i

,

m

=

∏

j

=

1

ω

⁢

(

m

)

q

j

b

j

,

where the

p

i

and

q

j

are all distinct primes, and the

a

i

and

b

j

are

positive integers

(not necessarily distinct), and

ω

⁢

(

x

)

is the

number of distinct prime factors function

, it becomes apparent that there are at least two different ways to sum up the prime factors.

The most obvious way is to simply test

∑

i

=

1

ω

⁢

(

n

)

p

i

=

∑

j

=

1

ω

⁢

(

m

)

q

j

,

in which case the first few Ruth-Aaron pairs are (5, 6), (24, 25), (49, 50), (77, 78), (104, 105), (153, 154), (369, 370), (492, 493), (714, 715), … (see A006145 in the OEIS).

The second way is to count repeated prime factors as they occur (e.g., the

sum of prime factors

of

72

=

2

3

⁢

3

2

is 12). Thus the test becomes

∑

i

=

1

ω

⁢

(

n

)

p

i

⁢

a

i

=

∑

j

=

1

ω

⁢

(

m

)

q

j

⁢

b

j

,

in which case the first few Ruth-Aaron pairs are (5, 6), (8, 9), (15, 16), (77, 78), (125, 126), (714, 715), (948, 949), … (see A039752 in the OEIS).

The pairs (5, 6), (77, 78), (714, 715), … work under either definition, and it can be observed that

|

μ

⁢

(

n

)

|

=

|

μ

⁢

(

m

)

|

=

1

, where

μ

⁢

(

x

)

is the Möbius function (at least for

n

<

1777028

).

Title

Ruth-Aaron pair

Canonical name

RuthAaronPair

Date of creation

2013-03-22 16:07:09

Last modified on

2013-03-22 16:07:09

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A51

Synonym

Ruth Aaron pair

Synonym

Aaron-Ruth pair

Synonym

Aaron Ruth pair

Defines

Ruth-Aaron number

Generated on Fri Feb 9 15:20:55 2018 by

LaTeXML
square-free number

square-free number

A

square-free

number is a

natural number

that contains no powers greater than 1 in its

prime factorization

. In other words, if

x

is our number, and

x

=

∏

i

=

1

r

p

i

a

i

is the prime factorization of

x

into

r

distinct primes, then

a

i

≥

2

is
always false for square-free

x

.

Note: we assume here that

x

itself must be greater than 1; hence 1 is not considered square-free. However, one must be alert to the particular

context

in which “square-free” is used as to whether this is considered the case.

The name derives from the fact that if any

a

i

were to be greater than or equal to two, we could be sure that at least one square divides

x

(namely,

p

i

2

.)

1

Asymptotic Analysis

The

asymptotic density

of square-free numbers is

6

π

2

which can be proved by application of a square-free variation of the

sieve of Eratosthenes

(

http://planetmath.org/SieveOfEratosthenes2

) as follows:

A

⁢

(

n

)

=

∑

k

≤

n

[

k

⁢

is squarefree

]

=

∑

k

≤

n

∑

d

2

|

k

μ

⁢

(

d

)

=

∑

d

≤

n

μ

⁢

(

d

)

⁢

∑

k

≤

n

d

2

|

n

1

=

∑

d

≤

n

μ

⁢

(

d

)

⁢

⌊

n

d

2

⌋

=

n

⁢

∑

d

≤

n

μ

⁢

(

d

)

d

2

+

O

⁢

(

n

)

=

n

⁢

∑

d

≥

1

μ

⁢

(

d

)

d

2

+

O

⁢

(

n

)

=

n

⁢

1

ζ

⁢

(

2

)

+

O

⁢

(

n

)

=

n

⁢

6

π

2

+

O

⁢

(

n

)

.

It was shown that the

Riemann Hypothesis

implies error term

O

⁢

(

n

7

/

22

+

ϵ

)

in the above

[

1

]

.

References

1

R. C. Baker and J. Pintz.

The distribution of square-free numbers.

Acta Arith.

, 46:73–79, 1985.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0535.10045

Zbl 0535.10045.

Title

square-free number

Canonical

name

SquarefreeNumber

Date of creation

2013-03-22 11:55:33

Last modified on

2013-03-22 11:55:33

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

17

Author

akrowne (2)

Entry type

Definition

Classification

msc 11A51

Synonym

square free number

Synonym

square free

Synonym

square-free

Synonym

squarefree

Related topic

MoebiusFunction

Related topic

SquareRootsOfRationals

Generated on Fri Feb 9 15:21:01 2018 by

LaTeXML
squarefull number

squarefull number

A natural number

n

is called squarefull (or powerful) if for every prime

p

|

n

we have

p

2

|

n

. In 1978 Erdős conjectured that we cannot have three consecutive squarefull natural numbers. If we assume the

ABC Conjecture

, there are only finitely many such consecutive triples.

Title

squarefull number

Canonical

name

SquarefullNumber

Date of creation

2013-03-22 11:45:26

Last modified on

2013-03-22 11:45:26

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

10

Author

KimJ (5)

Entry type

Definition

Classification

msc 11A51

Classification

msc 55-00

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Classification

msc 39A99

Classification

msc 39B72

Classification

msc 33E30

Synonym

powerful number

Related topic

NFullNumber

Generated on Fri Feb 9 15:21:03 2018 by

LaTeXML
Størmer number

Størmer number

A

Størmer number

or

arc-cotangent irreducible number

is a

positive integer

n

for which the

greatest prime factor

of

n

2

+

1

exceeds

2

⁢

n

. The first few Størmer numbers are 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 39, 40, 42, 44, 45, 48, 49, etc., listed in A005528 of Sloane’s OEIS. Weakening the

inequality

from

gpf

⁢

(

n

2

+

1

)

>

2

⁢

n

to

gpf

⁢

(

n

2

+

1

)

≥

2

⁢

n

makes no

difference

other than admitting 1 to the list (and possibly changing index offsets accordingly).

The Størmer numbers arise in connection with the problem of representing

Gregory numbers

t

a

b

as sums of Gregory numbers for

integers

. Conway and Guy explain in their book thus: “To find Størmer’s decomposition for

t

a

/

b

, you repeatedly multiply

a

+

b

⁢

i

by numbers

n

±

i

for which

n

is a Størmer number and the sign is chosen so that you can cancel the corresponding

prime number

p

(

n

is the smallest number for which

n

2

+

1

is

divisible

by

p

).”

Størmer numbers are named after the Norwegian physicist Carl Størmer (

http://planetmath.org/CarlStormer

).

References

1

John H. Conway & R. K. Guy,

The Book of Numbers

. New York: Copernicus Press (1996): 245 - 248.

2

J. Todd, “A problem on

arc tangent

relations”,

Amer. Math. Monthly

,

56

(1949): 517 - 528.

Title

Størmer number

Canonical name

StormerNumber

Date of creation

2013-03-22 17:52:16

Last modified on

2013-03-22 17:52:16

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Synonym

Stormer number

Synonym

Störmer number

Synonym

arc-cotangent irreducible number

Generated on Fri Feb 9 15:21:05 2018 by

LaTeXML
super-Poulet number

super-Poulet number

A

super-Poulet number

n

is a

Poulet number

which besides satisfying the

congruence

2

n

≡

2

mod

n

, each of its

divisors

d

i

(for

1

<

i

≤

τ

⁢

(

n

)

) also

satisfies

the congruence

2

d

i

≡

2

mod

d

i

.

Two examples: 341 is a super-Poulet number, with its divisors being 1, 11, 31 and 341 itself. We verify that

2

11

=

2048

=

11

×

186

+

2

and

2

31

=

2147483648

=

31

×

69273666

+

2

. 341 itself has already been checked when confirmed as a Poulet number. Now, 561 is a Poulet number but not a super-Poulet number since one of its divisors, 33, does not satisfy the congruence:

2

33

-

2

33

≈

260301048.18181818

⁢

…

.

The first few super-Poulet numbers are 341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, which are listed in A050217 of Sloane’s OEIS.

Title

super-Poulet number

Canonical name

SuperPouletNumber

Date of creation

2013-03-22 18:14:12

Last modified on

2013-03-22 18:14:12

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A51

Generated on Fri Feb 9 15:21:06 2018 by

LaTeXML
table of pseudoprimes below 2000 in bases 2 to 16

table of pseudoprimes below 2000 in bases 2 to 16

This table lists

pseudoprimes

(

http://planetmath.org/PseudoprimeP

)

p

of the form

b

p

-

1

≡

1

mod

b

below 2000.

Base

b

to base

b

OEIS reference

2

341, 561, 645, 1105, 1387, 1729, 1905

A001567

3

91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891

A005935

4

15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905

A020136

5

4, 124, 217, 561, 781, 1541, 1729, 1891

A005936

6

35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729

A005937

7

6, 25, 325, 561, 703, 817, 1105, 1825

A005938

8

9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949, 1001, 1105, 1281, 1365, 1387, 1417, 1541, 1649, 1661, 1729, 1785, 1905

A020137

9

4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288, 1387, 1541, 1729, 1891

A020138

10

9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729

A005939

11

10, 15, 70, 133, 190, 259, 305, 481, 645, 703, 793, 1105, 1330, 1729

A020139

12

65, 91, 133, 143, 145, 247, 377, 385, 703, 1045, 1099, 1105, 1649, 1729, 1885, 1891

A020140

13

4, 6, 12, 21, 85, 105, 231, 244, 276, 357, 427, 561, 1099, 1785, 1891

A020141

14

15, 39, 65, 195, 481, 561, 781, 793, 841, 985, 1105, 1111, 1541, 1891

A020142

15

14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921

A020143

16

15, 51, 85, 91, 255, 341, 435, 451, 561, 595, 645, 703, 1105, 1247, 1261, 1271, 1285, 1387, 1581, 1687, 1695, 1729, 1891, 1905

A020144

As you may have noticed, 1729 appears in all of these except for bases 7, 13 and 14, which share factors 7, 13 and 7, respectively, with 1729.

Title

table of pseudoprimes below 2000 in bases 2 to 16

Canonical name

TableOfPseudoprimesBelow2000InBases2To16

Date of creation

2013-03-22 16:50:15

Last modified on

2013-03-22 16:50:15

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A51

Generated on Fri Feb 9 15:21:08 2018 by

LaTeXML
table of small Knödel numbers K_n for 0<n<26

table of small Knödel numbers

K

n

for

0

<

n

<

26

The following table lists small Knödel numbers

K

n

for

0

<

n

<

26

. I’ve been deliberately arbitrary about what is small: for some

n

, especially even

n

, there are plenty of Knödel numbers to choose from so I only go up to 100; while for some odd bases there are so few I go up to 1000. And of course for

n

=

1

, the

K

n

are the

Carmichael numbers

, of which there is only one less than a thousand: 561.

n

m

∈

K

n

1

561, 1105, 1729, 2465, 2821, 6601, 8911

2

4, 6, 8, 10, 12, 14, 22, 24, 26, 30, 34, 38, 46, 56, 58, 62, 74, 82, 86, 94

3

9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93

4

6, 8, 12, 16, 20, 24, 28, 40, 44, 48, 52, 60, 68, 76, 80, 92

5

25, 65, 85, 145, 165, 185, 205, 265, 305, 365, 445, 485, 505, 545, 565, 685, 745, 785, 825, 865, 905, 965, 985

6

8, 10, 12, 18, 24, 30, 36, 42, 66, 72, 78, 84, 90

7

9, 15, 49, 91, 133, 217, 259, 301, 427, 469, 511, 553, 679, 721, 763, 889, 973

8

10, 12, 14, 16, 20, 24, 32, 40, 48, 56, 60, 80, 88, 96

9

21, 27, 45, 63, 99, 105, 117, 153, 171, 189, 207, 261, 273, 279, 333, 369, 387, 423, 429, 477, 513, 531, 549, 585, 603, 639, 657, 711, 747, 801, 873, 909, 927, 945, 963, 981

10

12, 24, 28, 30, 50, 70, 110, 130, 150, 170, 190, 230, 290, 310, 330, 370, 410, 430, 442, 470, 530, 532, 550, 590, 610, 670, 710, 730, 790, 830, 890, 910, 970

11

15, 35, 121, 341, 451, 455, 671, 781

12

14, 16, 18, 20, 22, 24, 36, 40, 42, 48, 60, 72, 80, 84

13

14, 15, 33, 169, 481, 793, 805, 949

14

15, 16, 18, 24, 26, 30, 44, 56, 98, 182, 264, 266, 392, 434, 510, 518, 602, 854, 938

15

16, 21, 39, 55, 63, 75, 195, 255, 275, 315, 435, 495, 555, 615, 795, 819, 915, 975

16

18, 20, 24, 28, 32, 40, 48, 52, 60, 64, 66, 80, 96

17

65, 77, 289, 665, 1649, 1921

18

20, 24, 30, 34, 36, 42, 54, 72, 78, 84, 88, 90

19

21, 51, 91, 361, 595, 703, 1387, 1955

20

22, 24, 38, 40, 48, 56, 60, 68, 80, 100

21

45, 57, 63, 85, 105, 117, 147, 231, 273, 357, 399, 441, 483, 585, 609, 651, 741, 777, 861, 903, 987

22

24, 28, 30, 70, 76, 102, 130, 132, 242, 682, 902, 910

23

25, 33, 35, 95, 119, 143, 455, 529

24

26, 30, 32, 36, 40, 42, 44, 46, 48, 60, 72, 80, 84, 96

25

27, 69, 125, 133, 165, 325, 385, 425, 725, 825, 925

Title

table of small Knödel numbers

K

n

for

0

<

n

<

26

Canonical name

TableOfSmallKnodelNumbersKnFor0N26

Date of creation

2013-03-22 17:17:25

Last modified on

2013-03-22 17:17:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A51

Generated on Fri Feb 9 15:21:10 2018 by

LaTeXML
table of Wilson quotient for 0<n<41

table of Wilson quotient for

0

<

n

<

41

The following table gives the

numerator

and denominator of the

Wilson quotient

for

0

<

n

<

41

, given in their lowest terms. If the denominator is 1, then

n

is prime, but if the denominator is the same as

n

, then

n

is

composite

and the numerator is simply

(

n

-

1

)

!

+

1

.

n

Numerator of

W

n

Denom.

1

2

1

2

1

1

3

1

1

4

7

4

5

5

1

6

121

6

7

103

1

8

5041

8

9

40321

9

10

362881

10

11

329891

1

12

39916801

12

13

36846277

1

14

6227020801

14

15

87178291201

15

16

1307674368001

16

17

1230752346353

1

18

355687428096001

18

19

336967037143579

1

20

121645100408832001

20

21

2432902008176640001

21

22

51090942171709440001

22

23

48869596859895986087

1

24

25852016738884976640001

24

25

620448401733239439360001

25

26

15511210043330985984000001

26

27

403291461126605635584000001

27

28

10888869450418352160768000001

28

29

10513391193507374500051862069

1

30

8841761993739701954543616000001

30

31

8556543864909388988268015483871

1

32

8222838654177922817725562880000001

32

33

263130836933693530167218012160000001

33

34

8683317618811886495518194401280000001

34

35

295232799039604140847618609643520000001

35

36

10333147966386144929666651337523200000001

36

37

10053873697024357228864849950022572972973

1

38

13763753091226345046315979581580902400000001

38

39

523022617466601111760007224100074291200000001

39

40

20397882081197443358640281739902897356800000001

40

Title

table of Wilson quotient for

0

<

n

<

41

Canonical name

TableOfWilsonQuotientFor0N41

Date of creation

2013-03-22 17:59:00

Last modified on

2013-03-22 17:59:00

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11A51

Classification

msc 11A41

Generated on Fri Feb 9 15:21:13 2018 by

LaTeXML