sign in sign up
<<
Home , Arithmetic function, Natural number, Perfect power

A for the of Perfect Powers

M A Nyblom

Scho ol of and Geospatial Science

RMIT University

GPO Box V Melb ourne

Victorial Australia

email michaelnyblomrmiteduau

Intro duction

n

A natural numb er of the form m where m is a p ositiveinteger and n is called a p erfect

power Unsolved problems concerning the of p erfect p owers ab ound throughout muchof

numb er The most famous of these is known as the Catalan conjecture which states

that the only p erfect p owers which dier by unity are the and It is of interest to

note that this particular problem has only recently b een solved using rather deep results from

the theory of cyclotomic elds see The set of p erfect p owers can naturally b e arranged

into an increasing sequence of distinct integers in which those p erfect powers expressible

with dieren texponents are treated as a single element of the sequence The rst few terms

of this sequence of p erfect p owers without duplication are

and is listed in the OnLine Encyclop edia of under Sloane A The

sequence in has many prop erties one b eing that the innite sum of its recipro cals is

convergent see a clear indication of the scarcity of the p erfect powers amongst the set

of natural numb ers This latter fact is naturally reected in the well known result that the

sequence of p erfect p owers has zero asymptotic densitythat is if N x denotes the

of elements of less than a p ositivereal x then lim N xx In view of this result

x

one may question what is the precise nature of the growth rate of the counting function

N x in particular can an asymptotic estimate for N x b e found We shall establish sucha

p

x as x distributional result for the sequence of p erfect p owers by proving that N x

As will b e seen this asymptotic formula can b e interpreted as stating that the p erfect squares

dominate the count of the sequence elements in as x To contrast the main result we

shall in develop a closedform expression for N x using elementary sieve metho ds

As will be seen this formula is some what reminiscent to Legendres counting function for

p

x x In what follows we denote the integer part of the number of primes in the interval

x by bxc

An Asymptotic Formula

To help establish the main results of this pap er we shall rst need to formally intro duce the

following family of sets

Denition Suppose x and n N nf g thenletA x denote the set of perfect powers

n

n n

having exponent n and which are less than or equal to x that is A xfk k N k xg

n

Wenow establish the asymptotic formula for the counting function N x

Theorem If N x denotes the number of sequence elements of that are less than or

p

equal to x then N x x as x

Pro of The rst step of the argument will b e to obtain upp er and lower functional b ounds

for N x Assuming without loss of generality that x observe A x for each

n

n N nf g but for n suciently large A xnfg Dening the auxiliary function

n

M xmaxfn N nf g A xnfg g we clearly see M x as A xnfg and

n

S

M x

that N x is equal to the number of elements of the set A A x Furthermore from

n

n

blog xc blog xc

the inequality x it is immediately deduced that M x blog xc

blog xc

Since for large x the family of sets fA xg are not mutually disjoint it follows that

n

n

blog xc

X

N xjAj jA xj

n

n

and since A x A one also has

jA xjjAj N x

n n

Now as A x there must exist a largest integer m such that m xm

n

p

n

xm that By taking the nth ro ot through the previous inequalitywe deduce m

p p

n n

is m b xc and so A xmust contain b xc elements Consequently and together

n

yields that

blog xc

X

p p

n

xcN x b xc b

n

Using the upp er and lower b ounds in we can establish required the asymptotic estimate

p

x observe for large x the following train of inequalities for N x as follows Dividing by

p p p p p

blog xc blog xc

n n

X X

b xc N x b xc xc b xc x b

p p p p p p

x x x x x x

n n

p p

blog xc



X

xc x b

p p

x x

n

p

b blog xc xc

p p



x x

Via an application of LHopitals rule it is easily seen that

log x blog xc

p p

 

x x

as x moreover by recalling lim bxcx we nally deduce from that

x

p

N x x asx

p

Remark Since the number of perfect squares less than or equal to x is given by b xc

p p

xc x we can interpret as stating that the perfect squares dominate and as b

the count of the sequence elements of as x

An Exact Formula

One of the earliest known sieve metho ds was a simple eective pro cedure for nding all prime

numb ers up to a certain bound x This pro cedure whichinvolves the systematic deletion of

p

all multiples of primes less than or equal to x was captured succinctly by Legendre using

a theoretical analog of the sifting pro cess known to day as the InclusionExclusion Principal

to study the prime counting function x jfp x p a prime gj His metho d led to an

p

exact formula for the number of primes in the interval x x in particular if denotes

the Mobius function then

k j

X

p

x

x d x

d

djP

x

Q

p

where the sum is taken over all of P p see pg In this section we

x

p x

shall employ the same elementary sieve metho d of Legendre to establish an exact formula

for the counting function N x which is similar in form to We b egin with a technical

lemma for the sets of Denition

Lemma For any set of m positive integers fn n g al l greater than unity

m

m

A xA x

n

n n

i

m



i

where n n denotes the of the m integers n n

m m

g Pro of We begin by demonstrating that A x A xA x for any n m N nf

n m

nm

which is the base step of our inductive argument Nowsincenjn mandmjn manynumber

nm

of the form k where k N can b e rewritten as a p erfect p ower having an exp onent n and

n m

m thus A x A x A x Let s A x A x with s then s k k

n m n m

nm

nm

for some k k N nf g W e have to pro duce a k N nf g such that s k As

n m



r

k k both k and k must have the same prime divisors Writing k p p p

r

r



n m

and k p p p we deduce from the equality k k that n m for each

r

i i

i r Consequently njn and mjn and so n n m for some N Thus

i i i i i

r



nm

s k where k p p p which establishes that A x A x A x

r

n m

nm

Now supp ose for m the set identity in holds for an arbitrary set of m p ositiveintegers

n n n n n n observe fn n g all greater than unity Then as

m m m m

from the inductive assumption and the base step that

m m

A x A x A x A x A x

n n n n

n n

m m

i i

m



i i

A x

n n n

m

 m

A x

n n

 m

Hence holds for m arbitrary p ositiv eintegers greater than unity and so the result is

established by the principal of

Theorem If x then the counting function for the sequence in is given by the

explicit expression

X



d

N xbxc dbx c

djP

x

Q

where the sum is taken over al l divisors of P p

x

pblog xc

Pro of We b egin by establishing a slight reformulation for the set A of Theorem Recall

blog xc

ing that A A x we claim if p p are the rst m primes less than or equal

n m

n

to blog xc then in fact A B where

m

x B A

p

r

r

x is included in the The inclusion B A follows automatically by denition as eachsetA

p

k

of sets which form A To establish the reverse inclusion A B rst observe that as

p p represent the complete list of primes less than or equal to blog xc every integer

m

n f blog xcg must b e divisible by at least one of these primes since otherwise by

log xc and the fundamental theorem of n would be divisible by a prime p b

n

so n blog xc a contradiction Consequentlyifgiven any s A x then s k and one

n

p

r

x and so A may write n p for some r f mg and N Thus s k

p r

r

every elementofA is contained in the set B

m

Now N xjAj jB j and since for x large the family of sets fA x g are not mutually

p

i

i

disjointwe deduce from an application of the InclusionExclusion Principal applied to the set

B that

m

X X

k

N x jA x A xj

p p

i i



k

i i m

k



k

where the expression i i m indicates that the sum is taken over all ordered

k

k element fi i g of the set f mg As the least common multiple of the

k

k prime num bers p p is clearly the pro duct d p p p observe from Lemma

i i i i i

 

k k



d

c noting here wehave again used the fact that jA x A xj jA xj bx

p p

d

i i



k

Q

p

n

that the number of elements in the set A x is b xc Dening P p we see

n x

pblog xc

m

that for each k f mg the inner in consists of adding terms of

k



d

c where d p p p is a of P having k distinct prime factors the form bx

i i i x



k

k

Consequently as p p p the double summation in must sum terms of the

i i i



k



d

form dbx c over all divisors dwhere d of P excluding d Finally by recalling that

x

we deduce that the right hand side of reduces to the right hand side of

Remark An immediate consequence of Theorem is that the number of nonperfect

P



d

c dbx powers less than or equal to x is equal to

djP

x

Numerical Example

We examine nowhow the explicit expression for N x in can b e practically implemented

to compute the numb er of p erfect p owers less than or equal to a given large p ositive real x

For notational convenience let the inner summation of b e denoted by

k j

X



p p

i i



k

S x x

k

i i m



k



p p

i i



k

Observe that in to evaluate each S x one must sum the terms bx c over

k

those subscripts i i whose values are chosen from the ordered k element subsets

k

m

of f mg consequently the numb er of summands is Thus on rst acquaintance

k

olve having to determine for each it would app ear that the calculation of S x would inv

k

m

k mall combinations of prime numb ers from the set fp p g However for

m

k

suciently large x this may not b e necessary since for certain values of k one can showthat

m

S x as follows

k

k

To b egin consider for any x the k xminfk N p p p xg

k

where again p denotes the ith We wish to rst show that if there are m

i

primes less than or equal to blog xc then k blog xc will b e at most m when m

Recalling for any n there exists a prime strictly b etween n and n Bertrands Postulate

observe as each p that

i

p p p p p p p p p p blog xc

m m m m m m m m m m

us when m we have p p blog xc and so k blog xc m Now for Th

m

m

blog xc p and k k blog xc we note that in the summation S x all

k

k

combinations of pro ducts p p p p blog xc Consequently from the

i i

k blog xc



k

blog xc blog xc

inequality x it is immediate that

  

blog xcp p p p blog xcp p

i i i i i i

  

k k k

x



m

p p

i i



k

Thus bx c and so the summation S terms all of xmust consist of adding

k

k

m

p



which are identically that is S x Hence for x the number of p erfect

k

k

powers less than or equal to x can b e calculated by the alternate expression

k blog xc

m

X X

m

k k

N x S x

k

k

k

k k blog xc

For x the value of the arithmetic function k blog xc will in practice b e much smaller

than the numb er of primes less than or equal to blog xc consequently in calculating N x

we shall only have to evaluate S x for the few values of k k blog xc In what

k

follows the reader may wish to consult the table of p erfect powers less than or equal to

by Serhart Sevki Dincer in

owers one can by Example Consider x From the table of perfect p

inspection deduce that N x To demonstrate the use of we shal l apply the alternate

expression in to verify the number of perfect powers less than or equal to x is Now

blog xc and so there are m primes namely less than blog xc

As we have that k blog xc and so from

X

k

N xS x S x

k

k

Using a calculator one nds in this instance that

p p p p p p p

     

S x b c b c b c b c b c b c b c

To evaluate S x rst recal l from denition

X



p p

i i



S x b c

i i





p p

i i



combinations of products c However of the Now if p p then b

i i



p p with i i the only products less than are and Thus

i i



the summation S x wil l consist of adding terms al l of which are identical ly

p p p p

   

together with the sum of the terms b c b c b c and b c which are

and respectively Consequently S x and so nal ly adding in the

alternating sum of binomial coecients in yields

N x

as required

References

S S Dincer httpwwwugb ccbilkentedutrsevkip owers up to lehtml

G Tenenbaum and M France The Prime and Their Distribution Student

Mathematics Library Volume AMS

H W Gould Problem H Fibonacci QuarterlyVol pg

P Mihailescu A class numb er free criterion for Catalans conjecture J