aarjaneiro/planetmath huggingface dataset:
Euler phi function

Euler phi function

For any

positive integer

n

,

φ

⁢

(

n

)

is the number of positive integers less than or equal to

n

which are

coprime

to

n

. The function

φ

is known as the

φ

function

. This function may also be denoted by

ϕ

.

Among the most useful of

φ

are the facts that it is

multiplicative

(meaning if

gcd

⁡

(

a

,

b

)

=

1

, then

φ

⁢

(

a

⁢

b

)

=

φ

⁢

(

a

)

⁢

φ

⁢

(

b

)

) and that

φ

⁢

(

p

k

)

=

p

k

-

1

⁢

(

p

-

1

)

for any prime

p

and any positive integer

k

. These two facts combined give a numeric computation of

φ

for all positive integers:

φ

⁢

(

n

)

=

n

⁢

∏

p

|

n

(

1

-

1

p

)

.

(1)

For example,

φ

⁢

(

2000

)

=

2000

⁢

∏

p

|

2000

(

1

-

1

p

)

=

2000

⁢

(

1

-

1

2

)

⁢

(

1

-

1

5

)

=

2000

⁢

(

1

2

)

⁢

(

4

5

)

=

8000

10

=

800

.

From

equation

(

1

), it is clear that

φ

⁢

(

n

)

≤

n

for any positive integer

n

with equality holding exactly when

n

=

1

. This is because

∏

p

|

n

(

1

-

1

p

)

≤

1

,

with equality holding exactly when

n

=

1

.

Another important fact about the

φ

function is that

∑

d

|

n

φ

⁢

(

d

)

=

n

,

where the sum extends over all

positive

divisors

of

n

. Also, by

definition

,

φ

⁢

(

n

)

is the number of units in the ring

ℤ

/

n

⁢

ℤ

of

integers

modulo

n

.

The

sequence

{

φ

⁢

(

n

)

}

appears in the OEIS as sequence

http://www.research.att.com/ njas/sequences/A000010

A000010.

Title

Euler phi function

Canonical

name

EulerPhiFunction

Date of creation

2013-03-22 11:45:11

Last modified on

2013-03-22 11:45:11

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

19

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11A25

Classification

msc 11-00

Classification

msc 14F35

Classification

msc 14H30

Classification

msc 20F34

Synonym

Euler totient function

Synonym

Euler

φ

function

Synonym

Euler

ϕ

function

Related topic

ProofThatEulerPhiIsAMultiplicativeFunction

Related topic

ValuesOfNForWhichVarphintaun

Related topic

PrimeResidueClass

Related topic

EulerPhiAtAProduct

Related topic

SummatoryFunctionOfArithmeticFunction

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

LaTeXML
every positive integer greater than 30 has at least one composite totative

every positive integer greater than 30 has at least one composite totative

Proposition

.

Every

positive integer

greater

than 30 has at least one

composite

totative

.

Proof.

Suppose we are given a positive integer

n

which is greater than 30.
Let

p

be the smallest

prime number

which does not divide

n

. Hence

gcd

⁡

(

n

,

p

2

)

=

1

. If

n

≤

50

, then

p

<

7

, so

p

2

≤

25

<

n

.
But if

n

>

50

and

p

≤

7

, then

p

2

<

50

<

n

. In either case we
get that

p

2

is a composite totative of

n

.

So now suppose

p

>

7

. Then

p

=

p

k

for some

k

>

4

. To

complete

the proof, it is enough to show that

p

2

is strictly smaller than
the

primorial

(

k

-

1

)

⁢

#

=

p

1

⁢

p

2

⁢

⋯

⁢

p

k

-

1

, which by

assumption

divides

n

. For then we would have

gcd

⁡

(

n

,

p

2

)

=

1

and

p

2

<

n

,
showing that

p

2

is a composite totative of

n

.

We now prove by

induction

that for any

k

>

4

, the

inequality

p

k

2

<

(

k

-

1

)

⁢

#

holds. For the

base case

k

=

5

we need to verify that

p

5

2

=

121

<

210

=

4

⁢

#

.

Now suppose

p

k

2

<

(

k

-

1

)

⁢

#

for some

k

>

4

. By Bertrand’s

postulate

,

p

k

+

1

<

2

⁢

p

k

, so applying the induction assumption, we
get that

p

k

+

1

2

<

4

⁢

p

k

2

<

4

⁢

(

k

-

1

)

⁢

#

.

But

4

<

k

<

p

k

, so

p

k

+

1

2

<

k

⁢

#

as desired.
∎

Title

every positive integer greater than 30 has at least one composite totative

Canonical name

EveryPositiveIntegerGreaterThan30HasAtLeastOneCompositeTotative

Date of creation

2013-03-22 16:58:19

Last modified on

2013-03-22 16:58:19

Owner

mps (409)

Last modified by

mps (409)

Numerical id

7

Author

mps (409)

Entry type

Result

Classification

msc 11A25

Related topic

SmallIntegersThatAreOrMightBeTheLargestOfTheirKind

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

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examples of perfect totient numbers

examples of perfect totient numbers

As has been proven, all powers of 3 are

perfect totient numbers

. Thus 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, etc. are all

examples of perfect totient numbers

.

Digging a little deeper, it follows from Perez Cacho’s proof that

3

⁢

p

is a perfect totient number when the prime is of the form

4

⁢

n

+

1

, where

n

is a smaller perfect totient number, that perfect totient numbers can be chained together in a somewhat similar manner to

Sophie Germain primes

and

safe primes

in

Cunningham chains

. That is, if

n

is a perfect totient number and

4

⁢

n

+

1

is a prime, then

12

⁢

n

+

3

is a perfect totient number. Although 1 is technically not a perfect totient number, since it is technically an

integer

power of 3, if we indulge in granting it perfect totient number status for the purpose of seeking chains, we find that it generates the chain 1, 15, 183, 2199.

The actual powers of 3 that are in fact perfect totient numbers disappoint when it comes to generating chains: 3, 39, 471; 9, 111; 27, 327; we need not even consider those

3

x

≡

1

mod

5

because in that case then clearly

5

|

(

4

⋅

3

x

+

1

)

; 243; 729, 8751; etc.

We’d be led astray, however, if we thought that we could generate a

complete

listing of perfect totient numbers simply by trying to make chains from powers of 3; such a method would overlook 4375 (which alas, however, does not make a chain). Neither does 4190263, 3932935775 nor 4764161215, three other perfect totient numbers that are not

divisible

by 3 and which do not give primes when multiplied by 4 and added 1.

How about other

multiples

of 3? There is 255, 3063, 36759; 363, 4359; if nothing else among chain beginnings below a million.

Title

examples of perfect totient numbers

Canonical name

ExamplesOfPerfectTotientNumbers

Date of creation

2013-03-22 16:37:45

Last modified on

2013-03-22 16:37:45

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A25

Generated on Fri Feb 9 15:13:24 2018 by

LaTeXML
existence of square roots of non-negative real numbers

existence of square roots of non-negative real numbers

Theorem

.

Every non-negative

real number

has a

square root

.

Proof.

Let

x

≥

0

∈

ℝ

. If

x

=

0

then the result is

trivial

, so suppose

x

>

0

and define

S

=

{

y

∈

ℝ

:

y

>

0

⁢

and

⁢

y

2

<

x

}

.

S

is nonempty, for if

0

<

y

<

min

⁡

{

x

,

1

}

, then

y

2

<

y

<

x

, and

y

∈

S

.

S

is also bounded above, for if

y

>

max

⁡

{

x

,

1

}

, then

y

2

>

y

>

x

, so such a

y

is an

upper bound

of

S

. Thus

S

is nonempty and bounded, and hence has a

supremum

which we denote

L

. We will show that

L

2

=

x

. First suppose

L

2

<

x

. By the

Archimedean Principle

there exists some

n

∈

ℕ

such that

n

>

(

2

⁢

L

+

1

)

/

(

x

-

L

2

)

. Then we have

(

L

+

1

n

)

2

=

L

2

+

2

⁢

L

n

+

1

n

2

<

L

2

+

2

⁢

L

n

+

1

n

<

x

⁢

.

(1)

So

L

+

1

/

n

is a member of

S

strictly

greater

than

L

, contrary to

assumption

. Now suppose that

L

2

>

x

. Again by the Archimedean Principle there exists some

n

∈

ℕ

such that

1

/

n

<

(

L

2

-

x

)

/

2

⁢

L

and

1

/

n

<

L

. Then we have

(

L

-

1

n

)

2

=

L

2

-

2

⁢

L

n

+

1

n

2

>

L

2

-

2

⁢

L

n

>

x

⁢

.

(2)

But there must exist some

y

∈

S

such that

L

-

1

/

n

<

y

<

L

, which gives

x

<

(

L

-

1

/

n

)

2

<

y

2

, so that

y

∉

S

, a

contradiction

. Thus it must be that

L

2

=

x

.
∎

Title

existence of square roots of non-negative real numbers

Canonical name

ExistenceOfSquareRootsOfNonnegativeRealNumbers

Date of creation

2013-03-22 16:32:42

Last modified on

2013-03-22 16:32:42

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 11A25

Related topic

AxiomOfAnalysis

Related topic

ArchimedeanProperty

Related topic

Supremum

Related topic

ExistenceOfNthRoot

Generated on Fri Feb 9 15:13:26 2018 by

LaTeXML
formula for the convolution inverse of a completely multiplicative function

formula for the convolution inverse of a completely multiplicative function

Corollary 1

.

If

f

is a

completely multiplicative function

, then its convolution inverse is

f

⁢

μ

, where

μ

denotes the Möbius

function

.

Proof.

Recall the Möbius

inversion

formula

1

*

μ

=

ε

, where

ε

denotes the convolution identity function. Thus,

f

⁢

(

1

*

μ

)

=

f

⁢

ε

. Since

pointwise

multiplication

of a completely multiplicative function distributes over convolution (

http://planetmath.org/PropertyOfCompletelyMultiplicativeFunctions

),

(

f

⋅

1

)

*

(

f

⁢

μ

)

=

f

⁢

ε

. Note that, for all

natural numbers

n

,

f

⁢

(

n

)

⁢

1

⁢

(

n

)

=

f

⁢

(

n

)

⋅

1

=

f

⁢

(

n

)

and

f

⁢

(

n

)

⁢

ε

⁢

(

n

)

=

ε

⁢

(

n

)

. Thus,

f

*

(

f

⁢

μ

)

=

ε

. It follows that

f

⁢

μ

is the convolution inverse of

f

.
∎

Title

formula for the convolution inverse of a completely multiplicative function

Canonical name

FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction

Date of creation

2013-03-22 16:55:09

Last modified on

2013-03-22 16:55:09

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

5

Author

Wkbj79 (1863)

Entry type

Corollary

Classification

msc 11A25

Related topic

CriterionForAMultiplicativeFunctionToBeCompletelyMultiplicative

Generated on Fri Feb 9 15:13:29 2018 by

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impractical number

impractical number

An

impractical number

n

is an even

abundant number

that is not a

practical number

under the second definition. The first few impractical numbers are 70, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 350, 354, 366, 372, 402, 426, 438, 444, 474, 490, 492, 498; these are listed in A007621 of Sloane’s OEIS.

Title

impractical number

Canonical name

ImpracticalNumber

Date of creation

2013-03-22 18:07:06

Last modified on

2013-03-22 18:07:06

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Related topic

PracticalNumber2

Generated on Fri Feb 9 15:13:32 2018 by

LaTeXML
iterated sum of divisors function

iterated sum of divisors function

The

iterated sum of divisors function

σ

k

⁢

(

n

)

is

a

k

in the

recurrence relation

a

0

=

n

and

a

i

=

σ

⁢

(

a

i

-

1

)

for

i

>

0

, where

σ

⁢

(

x

)

is the

sum of divisors function

.

Since

n

itself is included in the set of its

divisors

, the

sequence

generated by repeated iterations is an

increasing sequence

(that is, in

ascending order

). For example, iterating the sum of divisors function for

n

=

2

gives the sequence 2, 3, 4, 7, 8, 15, etc. Erdős conjectured that there is a limit for

(

σ

k

⁢

(

n

)

)

1

k

as

k

approaches

infinity

.

References

1

R. K. Guy,

Unsolved Problems in

Number Theory

New York: Springer-Verlag 2004: B9

Title

iterated sum of divisors function

Canonical

name

IteratedSumOfDivisorsFunction

Date of creation

2013-03-22 17:03:36

Last modified on

2013-03-22 17:03:36

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Generated on Fri Feb 9 15:13:34 2018 by

LaTeXML
iterated totient function

iterated totient function

The

iterated totient function

ϕ

k

⁢

(

n

)

is

a

k

in the

recurrence relation

a

0

=

n

and

a

i

=

ϕ

⁢

(

a

i

-

1

)

for

i

>

0

, where

ϕ

⁢

(

x

)

is Euler’s

totient function

.

After enough iterations, the

function

eventually

hits 2 followed by an

infinite

trail of ones. Ianucci et al define the “class”

c

of

n

as the

integer

such that

ϕ

c

⁢

(

n

)

=

2

.

When the iterated totient function is summed thus:

∑

i

=

1

c

+

1

ϕ

i

⁢

(

n

)

it can be observed that just as

2

x

is a

quasiperfect number

when it comes to adding up

proper divisors

, it is also “quasiperfect” when adding up the iterated totient function. Quite unlike

regular

perfect numbers

,

3

x

(which are obviously odd) are “

perfect

” when adding up the iterated

totient

.

References

1

D. E. Ianucci, D. Moujie & G. L. Cohen, “On

Perfect Totient Numbers

”

Journal of

Integer Sequences

,

6

, 2003: 03.4.5

2

R. K. Guy,

Unsolved Problems in

Number Theory

New York: Springer-Verlag 2004: B42

Title

iterated totient function

Canonical

name

IteratedTotientFunction

Date of creation

2013-03-22 16:33:09

Last modified on

2013-03-22 16:33:09

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A25

Related topic

PerfectTotientNumber

Generated on Fri Feb 9 15:13:36 2018 by

LaTeXML
Mangoldt function

Mangoldt function

The

Mangoldt function

Λ

is defined by

Λ

⁢

(

n

)

=

{

ln

⁡

p

,

if

n

=

p

k

, where

p

is a prime and

k

is a natural number

≥

1

0

,

otherwise

The

Moebius Inversion

Formula leads to the

identity

Λ

⁢

(

n

)

=

∑

d

|

n

μ

⁢

(

n

/

d

)

⁢

ln

⁡

d

=

-

∑

d

|

n

μ

⁢

(

d

)

⁢

ln

⁡

d

.

Title

Mangoldt function

Canonical name

MangoldtFunction

Date of creation

2013-03-22 11:47:06

Last modified on

2013-03-22 11:47:06

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

10

Author

KimJ (5)

Entry type

Definition

Classification

msc 11A25

Classification

msc 18E30

Classification

msc 81-00

Synonym

von Mangoldt function

Generated on Fri Feb 9 15:13:37 2018 by

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Mertens conjecture

Mertens conjecture

Franz Mertens conjectured that

|

M

⁢

(

n

)

|

<

n

where the

Mertens function

is defined as

M

⁢

(

n

)

=

∑

i

=

1

n

μ

⁢

(

i

)

,

and

μ

is the Möbius

function

.

However, Herman J. J. te Riele and Andrew Odlyzko have proven that there exist

counterexamples

beyond

10

13

, but have yet to find one specific counterexample.

The

Mertens conjecture

is related to the

Riemann hypothesis

, since

M

⁢

(

x

)

=

O

⁢

(

x

1

2

)

is another way of stating the Riemann hypothesis.

Given the

Dirichlet series

of the

reciprocal

of the

Riemann zeta function

, we find that

1

ζ

⁢

(

s

)

=

∑

n

=

1

∞

μ

⁢

(

n

)

n

s

is true for

ℜ

⁡

(

s

)

>

1

. Rewriting as Stieltjes integral,

1

ζ

⁢

(

s

)

=

∫

0

∞

x

-

s

⁢

𝑑

M

suggests this

Mellin transform

:

1

s

⁢

ζ

⁢

(

s

)

=

{

ℳ

⁢

M

}

⁢

(

-

s

)

=

∫

0

∞

x

-

s

⁢

M

⁢

(

x

)

⁢

d

⁢

x

x

.

Then it follows that

M

⁢

(

x

)

=

1

2

⁢

π

⁢

i

⁢

∫

σ

-

i

⁢

s

σ

+

i

⁢

s

x

s

s

⁢

ζ

⁢

(

s

)

⁢

𝑑

s

for

1

2

<

σ

<

2

.

References

1

G. H. Hardy and S. Ramanujan,

Twelve Lectures on Subjects Suggested by His Life and Work

3rd ed. New York: Chelsea, p. 64 (1999)

2

A. M. Odlyzko and H. J. J. te Riele, “Disproof of the Mertens Conjecture.”

J. reine angew. Math.

357

, pp. 138 - 160 (1985)

Title

Mertens conjecture

Canonical

name

MertensConjecture

Date of creation

2013-03-22 16:04:25

Last modified on

2013-03-22 16:04:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

10

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11A25

Synonym

Mertens’ conjecture

Synonym

Mertens’s conjecture

Generated on Fri Feb 9 15:13:39 2018 by

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Mertens’ first theorem

Mertens’ first theorem

For any

real number

x

≥

2

we have

∑

p

≤

x

ln

⁡

p

p

=

ln

⁡

x

+

O

⁢

(

1

)

for all prime integers

p

.

Moreover, the term

O

⁢

(

1

)

arising in this formula lies in the

open interval

(

-

1

-

ln

⁡

4

,

ln

⁡

4

)

.

Title

Mertens’ first theorem

Canonical name

MertensFirstTheorem

Date of creation

2013-03-22 11:46:06

Last modified on

2013-03-22 11:46:06

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

8

Author

KimJ (5)

Entry type

Theorem

Classification

msc 11A25

Classification

msc 17B45

Classification

msc 17B66

Generated on Fri Feb 9 15:13:41 2018 by

LaTeXML
Mertens function

Mertens function

Given an

integer

n

>

0

,

M

⁢

(

n

)

=

∑

i

=

1

n

μ

⁢

(

i

)

where

μ

⁢

(

x

)

is the Möbius function.

There is no

x

such that

M

⁢

(

x

)

>

x

. Franz Mertens conjectured that there is no

x

such that

M

⁢

(

x

)

>

x

, but this was proven false later.

References

1

F. Mertens, “ÃÆber eine zahlentheoretische Funktion”

Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber

, IIa 106, 761-830 (1897)

2

A. M. Odlzyko and te Riele, “Disproof of the

Mertens Conjecture

”

Journal fÃÆÃÂ¼r die reine und angewandte Mathematik

, 357, 138-160 (1985)

.1

External link

http://primefan.tripod.com/Mertens2500.html

Möbius and Mertens Values For n = 1 to 2500

Title

Mertens function

Canonical name

MertensFunction

Date of creation

2013-03-22 15:51:11

Last modified on

2013-03-22 15:51:11

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

10

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A25

Synonym

Mertens’ function

Synonym

Mertens’s function

Generated on Fri Feb 9 15:13:42 2018 by

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method for representing rational numbers as sums of unit fractions using practical numbers

method for representing rational numbers as sums of unit fractions using practical numbers

Fibonacci

’s application for

practical numbers

n

was an algorithm to represent

proper fractions

m

n

(with

m

>

1

) as sums of

unit fractions

∑

d

i

n

, with the

d

i

being

divisors

of the practical number

n

. (By the way, there are infinitely many practical numbers which are also

Fibonacci numbers

). The method is:

1.

Reduce the fraction to lowest terms. If the numerator is then 1, we’re done.

2.

Rewrite

m

as a sum of divisors of

n

.

3.

Make those divisors of

n

that add up to

m

into the numerators of fractions with

n

as denominator.

4.

Reduce those fractions to lowest terms, thus obtaining the representation

m

n

=

∑

d

i

n

.

To illustrate the algorithm, let’s rewrite

37

42

as a sum of unit fractions. Since 42 is practical, success is guaranteed.

At the first step we can’t reduce this fraction because 37 is a

prime number

. So we go on to the second step, and represent 37 as 2 + 14 + 21. This gives us the fractions

2

42

+

14

42

+

21

42

,

which we then reduce to lowest terms:

1

21

+

1

3

+

1

2

,

giving us the desired unit fractions.

References

1

M. R. Heyworth, “More on panarithmic numbers”

New Zealand Math. Mag.

17

(1980): 28 - 34

2

Giuseppe Melfi, “A survey on practical numbers”

Rend. Sem. Mat. Univ. Pol. Torino

53

(1995): 347 - 359

Title

method for representing rational numbers as sums of unit fractions using practical numbers

Canonical name

MethodForRepresentingRationalNumbersAsSumsOfUnitFractionsUsingPracticalNumbers

Date of creation

2013-03-22 18:07:00

Last modified on

2013-03-22 18:07:00

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Algorithm

Classification

msc 11A25

Generated on Fri Feb 9 15:13:44 2018 by

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Möbius function

Möbius function

The

Möbius

function

of

number theory

is the function

μ

:

ℤ

+

→

{

-

1

,

0

,

1

}

defined by

μ

⁢

(

n

)

=

{

1

,

if

n

=

1

0

,

if

p

2

|

n

for some prime

p

(

-

1

)

r

,

if

n

=

p

1

⁢

p

2

⁢

⋯

⁢

p

r

, where the

p

i

are distinct primes.

In other words,

μ

⁢

(

n

)

=

0

if

n

is not a square-free integer, while

μ

⁢

(

n

)

=

(

-

1

)

r

if

n

is

square-free

with

r

prime factors

. The function

μ

is a

multiplicative function

, and obeys the identity

∑

d

|

n

μ

⁢

(

d

)

=

{

1

if

n

=

1

0

if

n

>

1

where

d

runs through the

positive

divisors

of

n

.

Title

Möbius function

Canonical

name

MobiusFunction

Date of creation

2013-03-22 11:47:03

Last modified on

2013-03-22 11:47:03

Owner

mps (409)

Last modified by

mps (409)

Numerical id

11

Author

mps (409)

Entry type

Definition

Classification

msc 11A25

Classification

msc 55-00

Classification

msc 55-01

Synonym

Moebius function

Related topic

SquareFreeNumber

Related topic

SumOfFracmunn

Related topic

MoebiusInversionFormula

Related topic

ConvolutionMethod

Generated on Fri Feb 9 15:13:46 2018 by

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Nicomachus’ theorem

Nicomachus’ theorem

Theorem (Nicomachus). The sum of the cubes of the first

n

integers

is equal to the square of the

n

th

triangular number

. To put it algebraically,

∑

i

=

1

n

i

3

=

(

n

2

+

n

2

)

2

.

Proof.

There are several formulas for the triangular numbers. Gauss figured out that to compute

∑

i

=

1

n

i

,

one can, instead of

summing

the numbers one by one, pair up the numbers thus:

1

+

n

,

2

+

(

n

-

1

)

,

3

+

(

n

-

2

)

, etc., and each of these sums has the same result, namely,

n

+

1

. Since there are

n

of these sums, carrying this all the way through to the end, we are in effect squaring

n

+

1

, which is

(

n

+

1

)

2

=

(

n

+

1

)

⁢

(

n

+

1

)

=

n

2

+

n

. But this is redundant, since it includes both

1

+

n

and

n

+

1

, both

2

+

(

n

-

1

)

and

(

n

-

1

)

+

2

, etc., in effect, each of these twice. Therefore,

n

2

+

n

2

=

∑

i

=

1

n

i

.

As Sir Charles Wheatstone proved, we can rewrite

i

3

as

∑

j

=

1

i

(

2

⁢

i

⁢

j

+

i

)

.

That sum can always be rewritten as a sum of odd terms, namely

∑

k

=

1

i

(

i

2

+

i

+

2

⁢

k

)

.

Thus, the sum of the first

n

cubes is in fact also

∑

i

=

0

n

-

1

(

2

⁢

i

+

1

)

.

The sum of the first

n

-

1

odd numbers

is

n

2

, and therefore

∑

i

=

1

n

i

3

=

(

∑

i

=

1

n

i

)

2

,

as the theorem states.
∎

For example, the sum of the first four cubes is 1 + 9 + 27 + 64 = 100. This is also equal to 1 + 3 + 5 + 7 + 9 + 11 + 13 + 17 + 19 = 100. The

square root

of 100 is 10, the fourth triangular number, and indeed 10 = 1 + 2 + 3 + 4.

Title

Nicomachus’ theorem

Canonical name

NicomachusTheorem

Date of creation

2013-03-22 18:07:12

Last modified on

2013-03-22 18:07:12

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 11A25

Related topic

CubeOfAnInteger

Generated on Fri Feb 9 15:13:55 2018 by

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noncototient

noncototient

An

integer

n

>

0

is called a

noncototient

if there is no solution
to

x

-

ϕ

⁢

(

x

)

=

n

, where

ϕ

⁢

(

x

)

is Euler’s

totient function

. The first few noncototients are 10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130 (listed in A005278 of Sloane’s OEIS).

Browkin and Schinzel proved in 1995 that there are infinitely many noncototients. What is still unknown is whether they are all even. Goldbach’s

conjecture

would seem to suggest that this is the case: given a

semiprime

p

⁢

q

, it follows that

p

⁢

q

-

ϕ

⁢

(

p

⁢

q

)

=

p

⁢

q

-

(

p

-

1

)

⁢

(

q

-

1

)

=

p

+

q

-

1

, an

odd number

if

2

<

p

≤

q

.

Title

noncototient

Canonical

name

Noncototient

Date of creation

2013-03-22 15:55:48

Last modified on

2013-03-22 15:55:48

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Related topic

Nontotient

Generated on Fri Feb 9 15:13:57 2018 by

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