aarjaneiro/planetmath huggingface dataset:
prime quadruplet conjecture

prime quadruplet conjecture

Conjecture.

(Hardy & Littlewood) There are infinitely many

prime quadruplets

.

As with

twin primes

, prime quadruplets generally become scarcer the higher one looks for them with the aid of the computer, yet they also display the same unevenness of

distribution

: there is only one prime quadruplet between 40000 and 50000, yet there are three between 70000 and 80000. While Euclid proved long ago that there are infinitely many primes, it is still not known whether there are also infinitely many prime quadruplets.

The question is related to the

twin prime conjecture

: proving the

prime quadruplet conjecture

would automatically prove the twin prime conjecture as well. However, disproving the prime quadruplet conjecture might not necessarily disprove the twin prime conjecture as well.

Hardy and Littlewood stated their conjecture in more general terms as the prime

k

-tuple conjecture, with the case of the prime quadruplets being for

k

=

4

.

Title

prime quadruplet conjecture

Canonical name

PrimeQuadrupletConjecture

Date of creation

2013-03-22 19:00:38

Last modified on

2013-03-22 19:00:38

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11N05

Generated on Fri Feb 9 15:38:46 2018 by

LaTeXML
prime triples conjecture

prime triples conjecture

There exists infinitely many triples

p

,

q

,

r

of

prime numbers

such that

P

=

p

⁢

q

+

q

⁢

r

+

r

⁢

p

is a prime.

Title

prime triples conjecture

Canonical

name

PrimeTriplesConjecture

Date of creation

2013-03-22 13:35:17

Last modified on

2013-03-22 13:35:17

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Conjecture

Classification

msc 11N05

Related topic

TwinPrimesTheNumberOfConjuncture

Generated on Fri Feb 9 15:38:47 2018 by

LaTeXML
Redmond-Sun conjecture

Redmond-Sun conjecture

Conjecture. (Stephen Redmond & Zhi-Wei Sun) Given

positive integers

x

and

y

, and

exponents

a

and

b

(with all these numbers being

greater

than 1), if

x

a

≠

y

b

, then between

x

a

and

y

b

there are always primes, with only the following ten exceptions:

1.

There are no primes between

2

3

and

3

2

.

2.

There are no primes between

5

2

and

3

3

.

3.

There are no primes between

2

5

and

6

2

.

4.

There are no primes between

11

2

and

5

3

.

5.

There are no primes between

3

7

and

13

3

.

6.

There are no primes between

5

5

and

56

2

.

7.

There are no primes between

181

2

and

2

15

.

8.

There are no primes between

43

3

and

282

2

.

9.

There are no primes between

46

3

and

312

2

.

10.

There are no primes between

22434

2

and

55

5

.

See A116086 in Sloane’s OEIS for a listing of the

perfect powers

beginning primeless ranges before the next perfect power. As of 2007, no further

counterexamples

have been found past

55

5

.

Title

Redmond-Sun conjecture

Canonical name

RedmondSunConjecture

Date of creation

2013-03-22 17:26:50

Last modified on

2013-03-22 17:26:50

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11N05

Generated on Fri Feb 9 15:38:49 2018 by

LaTeXML
sequence of bounded variation

sequence of bounded variation

The

sequence

a

1

,

a

2

,

a

3

,

…

(1)

of

complex numbers

is said to be

of

bounded variation

, iff it satisfies

∑

n

=

1

∞

|

a

n

-

a

n

+

1

|

<

∞

.

Cf.

function of bounded variation

. See also
contractive sequence (

http://planetmath.org/ContractiveSequence

).

Theorem.

Every sequence of bounded variation is

convergent

(

http://planetmath.org/ConvergentSequence

).

Proof.

Let’s have a sequence (1) of bounded variation.
When

m

<

n

, we form the

telescoping sum

a

m

-

a

n

=

∑

i

=

m

n

-

1

(

a

i

-

a

i

+

1

)

from which we see that

|

a

m

-

a

n

|

≦

∑

i

=

m

n

-

1

|

a

i

-

a

i

+

1

|

.

This

inequality

shows, by the

Cauchy criterion for convergence

of
series, that the sequence (1) is a

Cauchy sequence

and thus

converges

. □

One kind of sequences of bounded variation is formed by the

bounded

monotonic sequences

of

real numbers

(those sequences
are convergent, as is well known).  Indeed, if (1) is a bounded
and e.g.

monotonically nondecreasing

sequence, then

a

i

≦

a

i

+

1

for each

⁢

i

,

whence

∑

i

=

1

n

|

a

i

+

1

-

a

i

|

=

∑

i

=

1

n

(

a

i

+

1

-

a

i

)

=

a

n

+

1

-

a

1

.

(2)

The boundedness of (1) thus implies that the

partial sums

(2) of
the series

∑

i

=

1

∞

|

a

i

+

1

-

a

i

|

with nonnegative
terms are bounded.  Therefore the last series is convergent, i.e.
our sequence (1) is of bounded variarion.

References

1

Paul Loya

:

Amazing and Aesthetic
Aspects of

Analysis

: On the incredible

infinite

.  A Course in Undergraduate Analysis, Fall 2006.
Available in http://www.math.binghamton.edu/dennis/478.f07/EleAna.pdf

Title

sequence of bounded variation

Canonical name

SequenceOfBoundedVariation

Date of creation

2014-11-28 21:01:47

Last modified on

2014-11-28 21:01:47

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Generated on Fri Feb 9 15:38:52 2018 by

LaTeXML
spam

spam

Title

spam

Canonical

name

Spam

Date of creation

2015-12-31 16:13:51

Last modified on

2015-12-31 16:13:51

Owner

vinayets10 (4012171)

Last modified by

pahio (2872)

Numerical id

2

Author

vinayets10 (2872)

Entry type

Topic

Generated on Fri Feb 9 15:38:53 2018 by

LaTeXML
Stern prime

Stern prime

If for a given

prime number

q

there is no smaller prime

p

and nonzero

integer

b

such that

q

=

2

⁢

b

2

+

p

, then

q

is a

Stern prime

. These primes were first studied by

Moritz Abraham Stern

, in connection to a lesser known

conjecture

of Goldbach’s. Like other mathematicians of the time, Stern considered 1 to be a prime number. Thus his list of Stern primes read thus: 2, 17, 137, 227, 977, 1187, 1493. A century later the list has been amended to include 3 (as in A042978 of Sloane’s OEIS) but no terms larger than 1493 have been found. The larger of a

twin prime

is not a Stern prime.

Title

Stern prime

Canonical

name

SternPrime

Date of creation

2013-03-22 16:19:10

Last modified on

2013-03-22 16:19:10

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N05

Generated on Fri Feb 9 15:38:55 2018 by

LaTeXML
twin prime conjecture

twin prime conjecture

Two consecutive

odd numbers

which are both prime are called

twin primes

, e.g. 5 and 7, or 41 and 43, or 1,000,000,000,061 and 1,000,000,000,063. But is there an

infinite

number of twin primes ?

In 1849 de Polignac made the more general

conjecture

that for every

natural number

n

, there are infinitely many prime pairs which have a

distance

of

2

⁢

n

. The case

n

=

1

is the

twin prime conjecture

.

In 1940, Erdős showed that there is a

constant

c

<

1

and infinitely many primes

p

such that

q

-

p

<

c

⁢

ln

⁡

p

where

q

denotes the next prime after

p

.
This result was improved in 1986 by Maier; he showed that a constant

c

<

0.25

can be used. The constant

c

is called the

twin prime constant.

In 1966, Chen Jingrun showed that there are infinitely many primes

p

such that

p

+

2

is either a prime or a

semiprime

.

Title

twin prime conjecture

Canonical

name

TwinPrimeConjecture

Date of creation

2013-03-22 13:21:32

Last modified on

2013-03-22 13:21:32

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

11

Author

alozano (2414)

Entry type

Conjecture

Classification

msc 11N05

Related topic

PrimeTriplesConjecture

Related topic

BrunsConstant

Defines

twin prime constant

Defines

twin primes

Generated on Fri Feb 9 15:38:57 2018 by

LaTeXML
values of 4^n and ∏@⁢ERROR(\slimits)⁢@⁢@⁢@_{i=1}^{π⁢(n)}⁢p_i for 0<n<26

values of

4

n

and

∏

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

i

=

1

π

⁢

(

n

)

⁢

p

i

for

0

<

n

<

26

The following table is in of the statement that

4

x

>

∏

i

=

1

π

⁢

(

x

)

p

i

is always true for any nonnegative

x

.

n

4

n

∏

i

=

1

π

⁢

(

n

)

p

i

4

n

-

∏

i

=

1

π

⁢

(

n

)

p

i

1

4

1

3

2

16

2

14

3

64

6

58

4

256

6

250

5

1024

30

994

6

4096

30

4066

7

16384

210

16174

8

65536

210

65326

9

262144

210

261934

10

1048576

210

1048366

11

4194304

2310

4191994

12

16777216

2310

16774906

13

67108864

30030

67078834

14

268435456

30030

268405426

15

1073741824

30030

1073711794

16

4294967296

30030

4294937266

17

17179869184

510510

17179358674

18

68719476736

510510

68718966226

19

274877906944

9699690

274868207254

20

1099511627776

9699690

1099501928086

21

4398046511104

9699690

4398036811414

22

17592186044416

9699690

17592176344726

23

70368744177664

223092870

70368521084794

24

281474976710656

223092870

281474753617786

25

1125899906842624

223092870

1125899683749754

Title

values of

4

n

and

∏

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

i

=

1

π

⁢

(

n

)

⁢

p

i

for

0

<

n

<

26

Canonical name

ValuesOf4nAnddisplaystyleprodi1pinPiFor0N26

Date of creation

2013-03-22 17:04:51

Last modified on

2013-03-22 17:04:51

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11N05

Classification

msc 11A25

Classification

msc 11A41

Generated on Fri Feb 9 15:38:59 2018 by

LaTeXML
Dirichlet’s theorem on primes in arithmetic progressions

Dirichlet’s theorem on primes in arithmetic progressions

If

a

is a

positive integer

and

(

a

,

b

)

=

1

, with

b

an

integer

, then there are infinitely many primes of the form

a

⁢

n

+

b

, with

n

an integer.

Title

Dirichlet’s theorem on primes in arithmetic progressions

Canonical name

DirichletsTheoremOnPrimesInArithmeticProgressions

Date of creation

2013-03-22 12:20:55

Last modified on

2013-03-22 12:20:55

Owner

vitriol (148)

Last modified by

vitriol (148)

Numerical id

9

Author

vitriol (148)

Entry type

Theorem

Classification

msc 11N13

Generated on Fri Feb 9 15:39:00 2018 by

LaTeXML
Siegel’s theorem on primes in arithmetic progressions

Siegel’s theorem on primes in arithmetic progressions

Definition 1

For

x

>

0

real,

q

,

a

positive integers

, we define

π

⁢

(

x

;

q

,

a

)

=

∑

p

≤

x

p

≡

a

(

mod

q

)

p

⁢

prime

1

i.e. the number of primes not exceeding

x

that are

congruent

to

a

modulo

q

.

Then the following holds:

Theorem 1

(Siegel) For all

A

>

0

, there is some constant

c

=

c

⁢

(

A

)

>

0

such that

π

⁢

(

x

;

q

,

a

)

=

L

⁢

i

⁢

(

x

)

φ

⁢

(

q

)

+

O

⁡

(

x

⁢

exp

⁡

(

-

c

⁢

log

⁡

x

)

)

for every

1

≤

q

≤

(

log

⁡

x

)

A

with

gcd

⁡

(

q

,

a

)

=

1

.

Note that it follows from this theorem that the distribution of primes among

invertible

residue classes

mod

q

does not depend on the residue class - that is, primes are evenly distributed into such classes.

A form of Dirichlet’s theorem on primes in arithmetic progressions states that

π

⁢

(

x

;

q

,

a

)

π

⁢

(

x

)

∼

1

φ

⁢

(

q

)

This follows easily from on noting that

L

⁢

i

⁢

(

x

)

=

x

log

⁡

x

+

⋯

.

Title

Siegel’s theorem on primes in arithmetic progressions

Canonical name

SiegelsTheoremOnPrimesInArithmeticProgressions

Date of creation

2013-03-22 17:58:29

Last modified on

2013-03-22 17:58:29

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11N13

Classification

msc 11A41

Related topic

PrimeNumberTheorem

Generated on Fri Feb 9 15:39:03 2018 by

LaTeXML
special case of Dirichlet’s theorem on primes in arithmetic progressions

special case of Dirichlet’s theorem on primes in arithmetic progressions

The special case of Dirichlet’s

theorem

for primes in arithmetic progressions for primes

congruent

to

1

modulo

q

where

q

itself is a prime can be established by the following elegant modification of Euclid’s proof (

http://planetmath.org/ProofThatThereAreInfinitelyManyPrimes

).

Let

f

⁢

(

n

)

=

n

q

-

1

n

-

1

=

1

+

n

+

n

2

+

⋯

+

n

q

-

1

. Let

n

>

1

be an

integer

, and suppose

p

∣

f

(

n

)

. Then

n

q

≡

1

(

mod

p

)

which implies by Lagrange’s theorem that either

q

∣

p

-

1

or

n

≡

1

(

mod

p

)

. In other words, every

prime divisor

of

f

⁢

(

n

)

is congruent to

1

modulo

q

unless

n

is congruent to

1

modulo that

divisor

.

Suppose there are only finitely many primes that are congruent to

1

modulo

q

. Let

P

be twice their

product

. Note that

P

≡

2

(

mod

q

)

. Let

p

be any prime divisor of

f

⁢

(

P

)

. If

p

≡

1

(

mod

q

)

, then

p

∣

P

which contradicts

f

⁢

(

P

)

≡

1

(

mod

P

)

. Therefore, by the above

P

≡

1

(

mod

p

)

. Therefore

f

⁢

(

P

)

≡

1

+

P

+

P

2

+

⋯

+

P

q

-

1

≡

1

+

1

+

1

+

⋯

+

1

≡

q

(

mod

p

)

. Since

q

is prime, it follows that

p

=

q

.
Then

P

≡

1

(

mod

p

)

implies

P

≡

1

(

mod

q

)

. However, that is

inconsistent

with our

deduction

that

P

≡

2

(

mod

q

)

above.
Therefore the original assumption that there are only finitely many primes congruent to

1

modulo

q

is false.

References

1

Henryk Iwaniec and Emmanuel Kowalski.

Analytic Number Theory

, volume 53 of

AMS Colloquium
Publications

.

AMS, 2004.

Title

special case of Dirichlet’s theorem on primes in arithmetic progressions

Canonical name

SpecialCaseOfDirichletsTheoremOnPrimesInArithmeticProgressions

Date of creation

2013-03-22 14:35:38

Last modified on

2013-03-22 14:35:38

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

9

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11N13

Generated on Fri Feb 9 15:39:05 2018 by

LaTeXML
table of primes in arithmetic progressions per Dirichlet’s theorem

table of primes in arithmetic progressions per Dirichlet’s theorem

Dirichlet’s theorem on primes in arithmetic progressions tells us that given the

n

th prime

p

n

, there are infinitely many primes of the form

m

⁢

p

n

+

1

. Obviously, for

p

1

=

2

, the primes of that form are merely the odd primes. For the other primes,

m

has to be even, but not much else appears obvious.

The leftmost column of this table gives the

n

th prime, the second column from the left gives the smallest prime of the form

m

⁢

p

n

+

1

, the third column from the left gives the second smallest prime of that form, etc. Apart from the leftmost column, none of the columns contain a sequence in

ascending order

.

p

n

2

3

5

7

11

13

17

19

23

29

31

3

7

13

19

31

37

43

61

67

73

79

5

11

31

41

61

71

101

131

151

181

191

7

29

43

71

113

127

197

211

239

281

337

11

23

67

89

199

331

353

397

419

463

617

13

53

79

131

157

313

443

521

547

599

677

17

103

137

239

307

409

443

613

647

919

953

19

191

229

419

457

571

647

761

1103

1217

1483

23

47

139

277

461

599

691

829

967

1013

1151

29

59

233

349

523

929

1103

1277

1451

1567

1741

31

311

373

683

1117

1303

1427

1489

1613

1861

2357

37

149

223

593

1259

1481

1777

1999

2221

2591

2887

41

83

739

821

1231

1559

1723

2297

2543

2707

2789

43

173

431

947

1033

1291

1549

1721

1979

2237

2753

47

283

659

941

1129

1223

1693

1787

2069

2351

2539

53

107

743

1061

1697

2333

2969

3181

3499

3923

4241

59

709

827

1063

1181

1889

2243

2833

3187

3541

3659

61

367

733

977

1709

1831

2441

3539

4027

4271

4637

67

269

1609

1877

2011

3083

3217

4021

4289

4423

4691

71

569

853

1279

1847

2131

2273

2557

2699

4261

5113

73

293

439

877

1607

1753

3067

3359

3797

3943

4673

79

317

1423

2213

2371

2687

3319

3793

4583

5531

5689

83

167

499

997

1163

1993

2657

4483

4649

5147

5479

89

179

1069

2137

2671

3739

3917

4273

4451

5519

6053

97

389

971

1553

1747

3299

3881

4463

4657

5821

6791

Title

table of primes in arithmetic progressions per Dirichlet’s theorem

Canonical name

TableOfPrimesInArithmeticProgressionsPerDirichletsTheorem

Date of creation

2013-03-22 18:13:15

Last modified on

2013-03-22 18:13:15

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11N13

Generated on Fri Feb 9 15:39:07 2018 by

LaTeXML
there are an infinite number of primes ≡1⁢@⁢ERROR(\symoperators)⁢m⁢o⁢d⁢ERROR(\tmspace)+.1667⁢e⁢m⁢ERROR(\tmspace)+.1667⁢e⁢m⁢m

there are an infinite number of primes

≡

1

⁢

@

⁢

\symoperators

⁢

m

⁢

o

⁢

d

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

m

This article proves a special case of Dirichlet’s

theorem

, namely that for any

integer

m

>

1

, there are an

infinite

number of primes

p

≡

1

(

mod

m

)

.

Let

p

be an

odd prime

not dividing

m

, let

Φ

k

⁢

(

x

)

be the

k

th

cyclotomic polynomial

, and note that

x

m

-

1

=

Φ

m

⁢

(

x

)

⋅

∏

d

∣

m

d

<

m

Φ

d

⁢

(

x

)

If

a

∈

ℤ

with

p

∣

Φ

m

(

a

)

, then clearly

p

∣

a

m

-

1

and thus

gcd

⁡

(

a

,

p

)

=

1

. In fact, the order (

http://planetmath.org/OrderGroup

) of

a

mod

p

is precisely

m

, for if it were not, say

a

d

≡

1

(

mod

p

)

for

d

<

m

, then

a

would be a root

mod

p

of

Φ

d

⁢

(

x

)

and thus

x

m

-

1

would have

multiple roots

mod

p

, which is a

contradiction

. But then, by Fermat’s little theorem, we have

a

p

-

1

≡

1

(

mod

p

)

, so since

m

is the least integer with this property, we have

m

∣

p

-

1

so that

p

≡

1

(

mod

m

)

.

We have thus shown that if

p

∤

m

and

p

∣

Φ

m

(

a

)

, then

p

≡

1

(

mod

m

)

. The result then follows from the following claim: if

f

⁢

(

x

)

∈

ℤ

⁢

[

x

]

is any

polynomial

of degree at least one, then the factorizations of

f

⁢

(

1

)

,

f

⁢

(

2

)

,

f

⁢

(

3

)

,

…

contain infinitely many primes. The proof is to Euclid’s proof of the infinitude of primes. Assume not, and let

p

1

,

…

,

p

k

be all of the primes. Since

f

is nonconstant, choose

n

with

f

⁢

(

n

)

=

a

≠

0

. Then

f

⁢

(

n

+

a

⁢

p

1

⋅

p

2

⁢

⋯

⁢

p

k

⁢

x

)

is clearly

divisible

by

a

, so

g

⁢

(

x

)

=

a

-

1

⁢

f

⁢

(

n

+

a

⁢

p

1

⋅

p

2

⁢

⋯

⁢

p

k

⁢

x

)

∈

ℤ

⁢

[

x

]

, and

g

⁢

(

m

)

≡

1

(

mod

p

1

⁢

⋯

⁢

p

k

)

for each

m

∈

ℤ

.

g

is nonconstant, so choose

m

such that

g

⁢

(

m

)

≠

1

. Then

g

⁢

(

m

)

is clearly divisible by some prime other that the

p

i

and thus

f

⁢

(

n

+

a

⁢

p

1

⋅

p

2

⁢

⋯

⁢

p

k

⁢

x

)

is as well. Contradiction.

Thus the set

Φ

m

⁢

(

1

)

,

Φ

m

⁢

(

2

)

,

…

contains an infinite number of primes in their factorizations, only a finite number of which can divide

m

. The

remainder

must be primes

p

≡

1

(

mod

m

)

.

Title

there are an infinite number of primes

≡

1

⁢

@

⁢

\symoperators

⁢

m

⁢

o

⁢

d

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

m

Canonical name

ThereAreAnInfiniteNumberOfPrimesequiv1modM

Date of creation

2013-03-22 17:43:02

Last modified on

2013-03-22 17:43:02

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11N13

Related topic

SpecialCaseOfDirichletsTheoremOnPrimesInArithmeticProgressions

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

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there are an infinite number of primes ≡±1⁢@⁢(ERROR(\symoperators)⁢m⁢o⁢d⁢4)

there are an infinite number of primes

≡

±

1

⁢

@

⁢

(

\symoperators

⁢

m

⁢

o

⁢

d

⁢

4

)

Theorem 1

.

There are an

infinite

number of primes

congruent

to

3

(

mod

4

)

.

Proof.

Choose any prime

p

≡

3

(

mod

4

)

; we find a prime of that form that exceeds

p

.

N

=

(

2

2

⋅

3

⋅

5

⋅

7

⁢

⋯

⁢

p

)

-

1

Clearly

N

≡

3

(

mod

4

)

, and thus must have at least one

prime factor

that is also

≡

3

(

mod

4

)

. But

N

is not

divisible

by any prime less than or equal to

p

, so must be divisible by some prime congruent to

3

(

mod

4

)

that exceeds

p

.
∎

Theorem 2

.

There are an infinite number of primes congruent to

1

(

mod

4

)

.

Proof.

Given

N

>

1

, we find a prime

p

>

N

with

p

≡

1

(

mod

4

)

. Let

p

be the smallest (odd) prime factor of

(

N

!

)

2

+

1

; note that

p

>

N

. Now

(

N

!

)

2

≡

-

1

(

mod

p

)

and therefore

(

N

!

)

p

-

1

≡

(

-

1

)

(

p

-

1

)

/

2

(

mod

p

)

.

By Fermat’s little

theorem

,

(

N

!

)

p

-

1

≡

1

(

mod

p

)

, so we have

(

-

1

)

(

p

-

1

)

/

2

≡

1

(

mod

p

)

The left-hand side cannot be -1, since then

0

≡

2

(

mod

p

)

. Thus

(

-

1

)

(

p

-

1

)

/

2

=

1

and it follows that

p

≡

1

(

mod

4

)

.
∎

Note that the variant of Euclid’s proof of the infinitude of primes used in the proof of Theorem

1

does not work for Theorem

2

, since we cannot conclude that an

integer

≡

1

(

mod

4

)

has a factor of the same kind.

References

1

Apostol, T

Introduction

to

Analytic Number Theory

, Springer 1976.

Title

there are an infinite number of primes

≡

±

1

⁢

@

⁢

(

\symoperators

⁢

m

⁢

o

⁢

d

⁢

4

)

Canonical name

ThereAreAnInfiniteNumberOfPrimesequivpm1pmod4

Date of creation

2013-03-22 16:56:59

Last modified on

2013-03-22 16:56:59

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11N13

Generated on Fri Feb 9 15:39:12 2018 by

LaTeXML
power of an integer

power of an integer

Let

n

be a non zero

integer

of

absolute value

not equal to one. The

power

of

n

, written

P

⁢

(

n

)

is defined by :

P

⁢

(

n

)

=

log

⁡

|

n

|

log

⁡

rad

⁡

(

n

)

.

where

rad

⁡

(

n

)

is the

radical

of the integer

n

.

1

1

Since

|

n

|

≠

1

, we have

rad

⁡

(

n

)

≠

1

also, so the

denominator

will not be equal to zero

If

n

=

m

k

, then

P

⁢

(

n

)

=

k

⁢

P

⁢

(

m

)

; in particular, if

n

is a prime power,

n

=

p

k

, then

P

⁢

(

n

)

=

k

. This observation explains why the term “power” is used
for this

concept

. At the same time, it is worth pointing out that, in general,
the power of an integer will not itself be an integer. For instance,

P

⁢

(

12

)

=

log

⁡

12

log

⁡

rad

⁡

(

12

)

=

log

⁡

12

log

⁡

6

=

1.3868

⁢

…

Note that it doesn’t matter what base one uses to compute the logarithm (as long as one uses the same base to compute the logarithm on the numerator and in the denominator!) because, upon changing base, both numerator and denominator will be multiplied by the same factor.

Title

power of an integer

Canonical name

PowerOfAnInteger

Date of creation

2013-03-22 14:22:17

Last modified on

2013-03-22 14:22:17

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

18

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 11N25

Synonym

power

Related topic

RadicalOfAnInteger

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