aarjaneiro/planetmath huggingface dataset:
convexity conjecture

convexity conjecture

Conjecture

(Hardy & Littlewood).

Given

integers

x

≥

y

>

1

, it is never the case that

π

⁢

(

x

+

y

)

>

(

π

⁢

(

x

)

+

π

⁢

(

y

)

)

, where

π

⁢

(

x

)

is the

prime counting function

.

For example: There are 269 primes below 1729. There are 304840 primes below 4330747. If we add up these values of the prime counting function, we get 305109. This is more than

π

⁢

(

4330747

+

1729

)

=

304949

.

Crandall and Pomerance believe this conjecture to be false but also that any

counterexample

is way too large to be discovered today. If we limit ourselves to 100 for both

variables

,

n

=

π

⁢

(

x

+

y

)

-

(

π

⁢

(

x

)

+

π

⁢

(

y

)

)

tends to fall in the range

-

8

<

n

<

1

.

References

1

R. Crandall & C. Pomerance,

Prime Numbers

: A Computational Perspective

, Springer, NY, 2001: 1.2.4

Title

convexity conjecture

Canonical

name

ConvexityConjecture

Date of creation

2013-03-22 16:45:51

Last modified on

2013-03-22 16:45:51

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11A41

Synonym

Hardy-Littlewood convexity conjecture

Generated on Fri Feb 9 15:16:01 2018 by

LaTeXML
double Mersenne number

double Mersenne number

A

double Mersenne number

is a number of the form

2

2

p

-

1

-

1

. Put another way,

negative

one plus 2 raised to the power of a

Mersenne number

. The first few double Mersenne numbers (and the ones small enough to show here) are: 3, 7, 127, 170141183460469231731687303715884105727.

If a double Mersenne number is itself prime, then it is called a

double Mersenne prime

. Obviously its index is then a

Mersenne prime

. The four double Mersenne numbers listed above are all primes, but as of today,

2

170141183460469231731687303715884105727

-

1

is a

probable prime

, despite an intense effort to find factors. (According to the Prime Pages, if it’s

composite

, its least

prime factor

must be at least

5

×

10

51

.

Title

double Mersenne number

Canonical name

DoubleMersenneNumber

Date of creation

2013-03-22 17:38:50

Last modified on

2013-03-22 17:38:50

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:16:02 2018 by

LaTeXML
emirp

emirp

An

emirp

is a

prime number

that in a given base

b

reads like a different prime number backwards. The plural of emirp is

emirps

, but

semirp

is occasionally used by amateurs.

For example, in base 10, the number 37 read backwards is 73, both of these are prime. The backwards number must be different. A006567 of Sloane’s OEIS lists the first fifty base 10 emirps.

A

palindromic prime

is not an emirp, since it reads the same number both ways. Thus, in binary,

Mersenne primes

can not be emirps.

Title

emirp

Canonical name

Emirp

Date of creation

2013-03-22 17:06:54

Last modified on

2013-03-22 17:06:54

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Classification

msc 11A63

Defines

semirp

Generated on Fri Feb 9 15:16:04 2018 by

LaTeXML
Euclid-Mullin sequence

Euclid-Mullin sequence

The

Euclid-Mullin sequence

is a

sequence

of

prime numbers

starting with 2 in which each prime afterwards is the least

prime factor

of one more than the

product

of the previous terms. Or, to put it algebraically,

a

1

=

2

and

a

n

for

n

>

1

is the smallest prime dividing

1

+

∏

i

=

1

n

-

1

a

i

.

(Actually,

a

1

can be set to 3, 7 or 43, resulting in a sequence that is exactly the same except for the order of its first four terms).

So, the product of 2 is 2, and one more than that is 3, which is prime. 2 times 3 is 6, and one more than 6 is 7, which is also prime. The product of 2, 3 and 7 is 42, just below the prime 43. The product of 2, 3, 7 and 43, however, is

composite

, namely

1807

=

13

×

139

, and thus 13 is the next term of the sequence. The sequence continues 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, etc., and is listed in A000945 of Sloane’s OEIS. (The first four terms of this sequence are the same as Sylvester’s sequence).

It follows from Euclid’s proof that there are infinitely many prime numbers that the Euclid-Mullin sequence does not contain any repeated terms. What is not known, however, is whether this sequence contains all the primes, as it is clear that very small primes can follow much larger primes. 31 is the smallest prime whose membership in this sequence is in question, but only the first 43 terms are known as of 2007. To find the 44th term, many are working on

factoring

2784908410762794071887414394815651793559267258537102016323316
20642982062389901741890579963524423782637435949041666525000702723914662388812510
545494307250950777886431051612811386531.

References

1

R. K. Guy & R. Nowakowski, “Discovering primes with Euclid,”

Delta

5

(1975): 49 - 63

2

A. A. Mullin, “

Recursive function

theory,”

Bull. Amer. Math. Soc.

69

(1963): 737

Title

Euclid-Mullin sequence

Canonical

name

EuclidMullinSequence

Date of creation

2013-03-22 17:37:41

Last modified on

2013-03-22 17:37:41

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:16:05 2018 by

LaTeXML
Euclid’s proof of the infinitude of primes

Euclid’s proof of the infinitude of primes

If there were only a finite amount of primes then there would be some largest prime

p

. However

p

!

+

1

is not divisible by any number

1

<

n

≤

p

, since

p

!

is, so

p

!

+

1

cannot be factored by the primes we already know, but every integer greater than one is divisible by at least one prime, so there must be some prime greater than

p

by which

p

!

+

1

is divisible.

Actually Euclid did not use

p

!

for his proof but stated that if there were a finite list

p

1

,

…

,

p

n

of primes, then the number

p

1

⁢

⋯

⁢

p

n

+

1

is not divisible by any of these primes and thus either prime and not in the list or divisible by a prime not in the list.

Title

Euclid’s proof of the infinitude of primes

Canonical name

EuclidsProofOfTheInfinitudeOfPrimes

Date of creation

2013-03-22 12:44:07

Last modified on

2013-03-22 12:44:07

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

9

Author

mathwizard (128)

Entry type

Proof

Classification

msc 11A41

Generated on Fri Feb 9 15:16:07 2018 by

LaTeXML
Euler’s lucky number

Euler’s lucky number

A

prime number

p

is one of

Euler’s

lucky numbers

if

n

2

-

n

+

p

for each

0

<

n

<

p

is also a prime. Put another way, a

lucky number of Euler

’s plus the

n

th

oblong number

produces a list of primes

p

-long. There are only six of them: 2, 3, 5, 11, 17 and 41, these are listed in A014556 of Sloane’s OEIS.

41 is perhaps the most famous of these. We can verify that 2 + 41 is 43, a prime, that 47 is also prime, so are 53, 61, 71, 83, 97, and so on to 1601, giving a list of 41 primes. Predictably, 1681 is

divisible

by 41, being its square. For

n

>

p

the formula does not consistently give only

composites

or only primes.

Title

Euler’s lucky number

Canonical name

EulersLuckyNumber

Date of creation

2013-03-22 16:55:33

Last modified on

2013-03-22 16:55:33

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Synonym

lucky number of Euler

Synonym

Eulerian lucky number

Generated on Fri Feb 9 15:16:08 2018 by

LaTeXML
examples of Pollard’s p-1 algorithm on a few integers

examples of Pollard’s

p

-

1

algorithm on a few integers

Let’s try Pollard’s

p

-

1

method (

http://

planetmath

.org/PollardsP1Algorithm

) for our sample

integers

with

B

=

10

and

a

=

2

whenever

n

is odd. Then our

a

p

e

are 256, 512, 32 and 128 and thus for each

n

we simply test whether 256, 511, 31 or 127 share any

divisors

with

n

.

Take

n

=

2039189

. It turns out 256, 511, 31 and 127 are all

coprime

to 2039189. So let’s try a higher test cap,

B

=

100

. Our

a

p

e

then become 18446744073709551616, 2417851639229258349412352, 562949953421312 and 33554432, putting the application of this

algorithm

out of reach for most

scientific calculators

. But suffice it to say that 18446744073709551615, 2417851639229258349412351, 562949953421311 and 33554431 are all coprime to

n

. What do you say we try

a

=

4

and give up if it doesn’t pan out? 340282366920938463463374607431768211455 and 5846006549323611672814739330865132078623730171903 disappoint as the others did. Finally 316912650057057350374175801344 gives something other than the 1 we’ve gotten accustomed to: 43. Dividing 2039189 by 43 gives us 47423. By now it’s understandable if you don’t want to put 47423 through this algorithm, since even a primitive implementation of

trial division

can quickly tell us that this

composite

is 47 times 1009. Thus the factorization of 2039189 is

43

⋅

47

⋅

1009

.

I’ll spare you the details for

n

=

1411041

. Suffice it to say that the settings of

a

and

B

that worked just now for 2039189 succeed in giving us the laughably small factor 3 of 1411041 which trial division would have given up right away.

As it turns out, Pollard’s

p

-

1

method applied to 4393547637856664251490043044051018234292171475232959 is doomed to failure for any

1

<

a

<

1001

and test caps of 10, 100 and 1000.

Title

examples of Pollard’s

p

-

1

algorithm on a few integers

Canonical

name

ExamplesOfPollardsP1AlgorithmOnAFewIntegers

Date of creation

2013-03-22 16:44:25

Last modified on

2013-03-22 16:44:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A41

Generated on Fri Feb 9 15:16:10 2018 by

LaTeXML
examples of probable primes

examples of probable primes

To give an example of a

probable prime

relative to a base:

4

341233

-

3

341233

has passed preliminary

primality tests

relative to bases 2, 3, 5, 7, 11, 13 and 101. Its

square root

is approximately

2.3362

⋅

10

102721

, which makes a conclusive primality test by

trial division

in a reasonable time period impractical.

To give an example of a probable prime by a pattern: this pattern

2

2

-

1

=

3

,

2

3

-

1

=

7

,

2

7

-

1

=

127

2

127

-

1

=

170141183460469231731687303715884105727

suggests that

2

170141183460469231731687303715884105727

-

1

might be a

Mersenne prime

. But since this is larger than the largest known Mersenne prime

2

30402457

-

1

(as of 2005), a

Lucas-Lehmer test

might take longer than the

average

human lifetime.

On the other hand,

123456789

⋅

10

123456789

+

123456789

is not a probable prime, because even though it is much larger than either of the probable primes given above, it is clearly

divisible

by

3

2

.

Title

examples of probable primes

Canonical name

ExamplesOfProbablePrimes

Date of creation

2013-03-22 15:53:49

Last modified on

2013-03-22 15:53:49

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A41

Generated on Fri Feb 9 15:16:11 2018 by

LaTeXML
examples of the Fermat method on a few integers

examples of the Fermat method on a few integers

Take

n

=

2039183

. The

square root

is approximately 1428, so that’s what we set our iterator’s initial state to. The test cap is 339865.

At

i

=

1428

, we find that

1428

2

-

2039183

=

1

, clearly an

integer

.

Then,

1428

-

1

=

1427

and

1428

+

1

=

1429

. By

multiplication

, we verify that

2039183

=

1427

⋅

1429

, indeed. It is the

product

of a

twin prime

. By

trial division

, this would have taken 225 test

divisions

.

However, there are integers for which trial division performs much better than the

Fermat method

. For example, take

n

=

1411041

. Our iterator starts at 1188 and the test cap is 235175.

At

i

=

1188

, we find that

1188

2

-

1411041

≈

17.4068952

.

Similar

frustration at 1189, 1190, etc.

It’s not until we get to

i

=

235175

that we finally get

235175

2

-

1411041

=

235172

.

Then,

235175

-

235172

=

3

, and

235175

+

235172

=

470347

, and we verify that indeed

3

⋅

470347

=

2039183

. This took 233987

instances

of squaring,

subtraction

and square root extraction, which would have taken just 196 test divisions in trial division.

How about 4393547637856664251490043044051018234292171475232959? The square root is approximately 6628384145368058140794809 and the test cap is 732257939642777375248340507341836372382028579205495. At worst, the Fermat method could take as many as 732257939642777375248340500713452227013970438410686 instances of squaring, subtraction and square root extraction to give a result. Clearly a more sophisticated method is needed to factorize an integer of this magnitude.

Title

examples of the Fermat method on a few integers

Canonical

name

ExamplesOfTheFermatMethodOnAFewIntegers

Date of creation

2013-03-22 16:39:28

Last modified on

2013-03-22 16:39:28

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A41

Generated on Fri Feb 9 15:16:13 2018 by

LaTeXML
factorial prime

factorial prime

A

factorial prime

is a number that is one less or one more than a

factorial

and is also a

prime number

. The first few factorial primes are: 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199 (

sequence

A088054 in the OEIS). It is conjectured that only for

n

=

3

are both

n

!

-

1

and

n

!

+

1

both primes.

Factorial primes have a rôle in an argument that 1 is not a prime number. If

n

is a

positive integer

and

p

is a prime number,

n

!

+

p

is never a prime for

p

<

n

, because obviously it will be a

multiple

of

p

, just as

n

!

is. But

n

!

+

1

, even though it certainly is a multiple of 1, can be a prime, specifically, a factorial prime. (The same is also true if we subtract instead of add).

Title

factorial prime

Canonical name

FactorialPrime

Date of creation

2013-03-22 16:19:19

Last modified on

2013-03-22 16:19:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Classification

msc 05A10

Classification

msc 11B65

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

LaTeXML
Fermat method

Fermat method

The

Fermat method

is a factorization algorithm for

odd integers

that tries to represent them as the difference of two square numbers.

Call the Fermat method algorithm with an

integer

n

=

2

⁢

x

+

1

.

1.

Initialize the iterator

i

=

⌈

n

⌉

and test cap at

i

2

-

n

.

2.

Compute

j

=

i

2

-

n

.

3.

Test if

j

is an integer. If it is, break out of subroutine and return

i

-

j

.

4.

Increment iterator

i

once. If

i

<

n

+

9

6

, go to step 2.

5.

Return

n

.

For example,

n

=

221

. The

square root

is approximately 15, so that’s what we set our iterator’s initial state to. The test cap is 39.

At

i

=

15

, we find that

15

2

-

221

=

2

, clearly an integer.

Then,

15

-

2

=

13

and

15

+

2

=

17

. By

multiplication

, we verify that

221

=

13

⋅

17

, indeed. By

trial division

, this would have taken 15 test divisions.

However, there are integers for which trial division performs much better than the Fermat method, such as numbers of the form

3

⁢

p

where

p

is an

odd prime

greater than 3.

References

1

R. Crandall & C. Pomerance,

Prime Numbers

: A Computational Perspective

, Springer, NY, 2001: 5.1

Title

Fermat method

Canonical name

FermatMethod

Date of creation

2013-03-22 16:39:03

Last modified on

2013-03-22 16:39:03

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Algorithm

Classification

msc 11A41

Synonym

Fermat’s method

Synonym

Fermat factorization method

Synonym

Fermat’s factorization method

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

LaTeXML
Fortunate number

Fortunate number

Given a

positive integer

n

, the

n

th

Fortunate number

F

n

>

1

is the

difference

between the

primorial

∏

i

=

1

π

⁢

(

n

)

p

i

(where

π

⁢

(

x

)

is the

prime counting function

and

p

i

is the

i

th

prime number

) and the nearest prime number above (ignoring the primorial prime that may be there). For example, the 3rd Fortunate number is 7, since the third primorial is 30 since the next highest prime is 37 (the primorial prime 31 is ignored).

The first few Fortunate numbers are 3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, etc. listed in

http://www.research.att.com/ njas/

sequences

/A005235

A005235 in Sloane’s OEIS. Some Fortunate numbers occur more than once, such as 23, which occurs for both the fifth and eighth primorials. so, the inequality

F

n

>

n

always holds. These numbers are named after the anthropologist Reo Fortune, who conjectured on their

primality

.

Title

Fortunate number

Canonical

name

FortunateNumber

Date of creation

2013-03-22 17:31:10

Last modified on

2013-03-22 17:31:10

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:16:18 2018 by

LaTeXML
Fortunate prime

Fortunate prime

A

Fortunate prime

is a

Fortunate number

that is also a

prime number

. As of 2007, all known Fortunate numbers are Fortunate primes. Richard K. Guy and Eric W. Weisstein have verified that the first

thousand

Fortunate numbers are indeed Fortunate primes.

Title

Fortunate prime

Canonical

name

FortunatePrime

Date of creation

2013-03-22 17:31:13

Last modified on

2013-03-22 17:31:13

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

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

LaTeXML
Fortune’s conjecture

Fortune’s conjecture

(Reo F. Fortune) For any

integer

n

>

0

, the

difference

between the

primorial

n

⁢

#

=

∏

i

=

1

π

⁢

(

n

)

p

i

(where

π

⁢

(

x

)

is the

prime counting function

and

p

i

is the

i

th

prime number

) and the nearest prime number above (excluding the possible primorial prime

n

⁢

#

+

1

) is always a prime number. That is, any

Fortunate number

is a

Fortunate prime

.

It is

obvious

that since

n

⁢

#

is

divisible

by each prime

p

<

p

π

⁢

(

n

)

, then each

n

⁢

#

+

p

will also be divisible by that same

p

and thus not prime. If there is a prime

q

>

n

⁢

#

+

1

such that there is a

composite number

m

=

q

-

n

⁢

#

, then

m

would have to have at least two

prime factors

both of which would have to be divisible by primes greater than

p

π

⁢

(

n

)

.

Despite verification for the first

thousand

primorials, this

conjecture

remains unproven as of 2007. Disproof could require finding a composite Fortunate number. Such a number would have to be odd, and indeed not divisible by the first thousand primes. Chris Caldwell, writing for the Prime Pages, argues that by the

prime number theorem

, finding a composite Fortunate number is tantamount to searching for a

prime gap

at least

(

log

⁡

n

⁢

#

)

2

long immediately following a primorial, something he considers unlikely.

References

1

S. W. Golomb, “The evidence for Fortune’s conjecture,”

Math. Mag.

54

(1981): 209 - 210. MR 82i:10053

Title

Fortune’s conjecture

Canonical

name

FortunesConjecture

Date of creation

2013-03-22 17:31:17

Last modified on

2013-03-22 17:31:17

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11A41

Generated on Fri Feb 9 15:16:21 2018 by

LaTeXML
Fürstenberg’s proof of the infinitude of primes

Fürstenberg’s proof of the infinitude of primes

Fürstenberg’s proof (

[

1

]

,

[

2

]

) that there are infinitely many primes is an amusing and beautiful blend of

elementary number theory

and point-set topology.

Consider the

arithmetic progression

topology

on the

positive integers

, where a basis of open sets is given by subsets of the form

U

a

,

b

=

{

n

∈

ℤ

+

|

n

≡

b

mod

a

}

. Arithmetic progressions themselves are by

definition

open, and in fact clopen, since

U

a

,

b

c

=

⋃

c

≠

b

U

a

,

c

where the union is taken over a set of distinct

residue classes

modulo

a

. Hence the

complement

of

U

a

,

b

is a union of open sets and so is open, so

U

a

,

b

itself is closed (and hence clopen).

Consider the set

U

=

∪

p

U

p

,

0

, where the union runs over all primes

p

. Then the complement of

U

in

ℤ

+

is the single

element

{

1

}

, which is clearly not an open set (every open set is

infinite

in this topology). Thus

U

is not closed, but since we have written

U

as a union of

closed sets

and a

union of closed sets is again closed, this implies that there must be infinitely many terms appearing in that union, i.e. that there must be infinitely many distinct primes.

References

1

Furstenberg, Harry,

On the infinitude of primes

,
American Mathematical Monthly, Vol. 62, 1955, p. 353.

2

Ribenboim, Paulo.

The New Book of

Prime Number

Records

. Springer, 1996. p. 10

Title

Fürstenberg’s proof of the infinitude of primes

Canonical

name

FurstenbergsProofOfTheInfinitudeOfPrimes

Date of creation

2013-03-22 14:42:10

Last modified on

2013-03-22 14:42:10

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Proof

Classification

msc 11A41

Related topic

HausdorffSpaceNotCompletelyHausdorff

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

LaTeXML
Google number

Google number

The n-th

Google number

is the earliest n-digit

prime number

found as a substring in

decimal expansion

of e. Hence the first few Google numbers are 2, 271, 4523, 74713, 904523, 2718281, 72407663 … (

Sequence

A095935 of OEIS). They were named due to unusual hiring ad that Google put up on a billboard on July 29, 2004.

Title

Google number

Canonical

name

GoogleNumber

Date of creation

2013-03-22 19:20:42

Last modified on

2013-03-22 19:20:42

Owner

Kausthub (26471)

Last modified by

Kausthub (26471)

Numerical id

4

Author

Kausthub (26471)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:16:27 2018 by

LaTeXML