Asymptotic Formulae for the n-th Perfect Power

Transcription

1 Joural of Iteger Sequeces, Vol. 5 0, Article.5.5 Asymptotic Formulae for the -th Perfect Power Rafael Jakimczuk Divisió Matemática Uiversidad Nacioal de Lujá Bueos Aires Argetia jakimczu@mail.ulu.edu.ar I memory of my sister Fedra Maria Jakimczuk Abstract Let P be the -th perfect power. I this article we prove asymptotic formulae for P. For example, we prove the followig formula P = 5/3 7/ /3 9/7 + 6/5 3/ + o 3/. Itroductio A atural umber of the form m where m is a positive iteger ad is called a perfect power. The first few terms of the iteger sequece of perfect powers are, 4, 8, 9, 6, 5, 7, 3, 36, 49, 64, 8, 00,, 5, 8... ad they form sequece A00597 i Sloae s Ecyclopedia. Let P be the -th perfect power. That is, P =,P = 4,P 3 = 8,P 4 = 9,... I this article we prove asymptotic formulae for P. For example, P = 5/3 7/ /3 9/7 + 6/5 3/ + o 3/. This formula is a corollary of our mai theorem Theorem 6, which ca give as may terms i the expasio as desired.

2 There exist various theorems ad cojectures o the sequece P. For example, the followig theorem: = µk ζk = 0, P = k= where µk is the Möbius fuctio ad ζk is the Riema zeta fuctio. We also have the followig theorem called the Goldbach-Euler theorem: P =. = This result was first published by Euler i 737. Euler attributed the result to a letter ow lost from Goldbach. Mihăilescu [4, 5, 6] proved that the oly pair of cosecutive perfect powers is 8 ad 9, thus provig Catala s cojecture. The Pillai s cojecture establish the followig limit lim P + P =. This is a usolved problem. There exist algorithms for detectig perfect powers [, ]. Let Nx be the umber of perfect powers ot exceedig x. M. A. Nyblom [7] proved the followig asymptotic formula Nx x. M. A. Nyblom [8] also obtaied a formula for the exact value of Nx usig the iclusioexclusio priciple. Let be the h-th prime. Cosequetly we have, p =,p = 3,p 3 = 5,p 4 = 7,p 5 =,p 6 = 3,... Jakimczuk [3] proved the followig theorem where more precise formulae for Nx are established. This theorem will be used later. Theorem. Let be the h-th prime with h, where h is a arbitrary but fixed positive iteger. The h Nx = k+ k= i < <i k h p i p ik < p x i p ik + + ox /, where the ier sum is take over the k-elemet subsets {i,...,i k } of the set {,,...,h } such that the iequality p i p ik < holds. If h = 5 the Theorem becomes, Nx = x + 3 x + 5 x 6 x + 7 x 0 x + + o x. Note that equatio iclude the cases h =, 3, 4. I geeral, equatio for a certai value of h = k iclude the cases h =, 3,...,k. This fact is a direct cosequece of equatio.

3 Some Lemmas The followig lemma is a immediate cosequece of the biomial theorem. Lemma. We have + x α = + α + ox x 0, + x α = + αx + Ox x 0, + x α αα = + αx + x + Ox 3 x 0. Lemma 3. Let P be the -th perfect power. We have P. Proof. Equatio gives Nx x. Cosequetly NP = P. Therefore P. Lemma 4. Let be the h-th prime. If h 3 the we have Proof. We have 3 <. 3 < < 3 < 6. Clearly, the last iequality is true if h 5 sice 7. O the other had, we have p p 3 = 3 5 = 5 < 6 p 3 p 4 = 5 7 = 35 < 6 3 The Fudametal Lemma The followig lemma is a characterizatio of asymptotic formulae for P. The lemma prove the existece of asymptotic formulae for P. Lemma 5. Let h 3 be the h-th prime. We have m P = 5/3 + d i g i + + o +, 3 where > 5/3 > g > > g m > +, the d i are ratioal coefficiets ad i equatio 3 appear the terms + p i i =,...,h. Besides the ratioal expoets 5/3 ad g i i =,...,m are of the form b i c i where b i ad c i are relatively prime ad the c i are squarefree itegers with prime divisors bouded by. 3

4 Proof. We shall use mathematical iductio. First, we shall prove that the lemma is true for h = 3. If h = the Theorem becomes see Nx = x + + o 3 x. Substitutig x = P ito this equatio ad usig Lemma 3 we obtai NP = = P + + o 3 P = P + + o /3. That is P = + + o /3. Therefore P = + + o /3 If h = 3 the Theorem becomes see = + + o 5/3 + + o 4/3 = + + o 5/3. 4 Nx = x + 3 x + + o 5 x. Substitutig x = P ito this equatio ad usig equatio 4, Lemma 3 ad Lemma we obtai NP = = P / + P /3 + + o /5 = P / o 5/3 /3 + + o /5 = P / + /3 + + o /3 /3 + + o /5 = P / + /3 + /3 + o /3 + + o /5 = P / + /3 + + o /5. That is Therefore P / = /3 + + o /5. That is P = /3 + + o /5 = 5/3 + + o 7/5. P = 5/3 + + o 7/5. 5 Equatio 5 is Lemma 5 for h = 3. Cosequetly the lemma is true for h = 3. Suppose that the lemma is true for h 3. We shall prove that the lemma is also true for h 4. 4

5 We have see Nx = + h k+ k= i < <i k h p i p ik < h h x /p i + k+ i= k= p x i p ik + + ox / = x / i < <i k h p i p ik < x p i p ik + + ox / h 4. 6 Substitutig x = P ito 6 ad usig Lemma 3 we obtai h h = P / + P /p i + k+ i= By iductive hypothesis we have k= i < <i k h p i p ik < P p i p ik + + o / h 4. 7 P = 5/3 + a i r i + + o + h 4, 8 where > 5/3 > r > > r s > +, the a i are ratioal coefficiets ad i equatio 8 appear the terms + p i i =,...,h. Besides the ratioal expoets 5/3 ad r i i =,...,s are of the form l i f i where l i ad f i are relatively prime ad the f i are squarefree itegers with prime divisors bouded by. Equatio 8 gives P = /3 + a i +r i + + o +, 9 where Cosequetly /3 + a i +r i + + o + /3. /3 + a i +r i + + o + = O /3 = o. 0 5

6 Let t 3 be a positive iteger. Equatios 9, 0 ad Lemma give P /t = /t /3 + = /t + t /3 + a i +r i + + o + /t a i +r i + + o + + O /3 = /t t 3 + t + t a i +r i+ + + p + o + h t + O 3 + t t Note that if t 3 the see Lemma 4 t. ad if t 3 the t + p + o + h t = o h /p, O 3 + t = o h /p. 3 Cosequetly becomes see ad 3 P /t = /t t 3 + t + t a i +r i+ t + o /. 4 Note that see 4 if t 3 the expoet < ad cosequetly also + < ad t 3 t + r i + < sice + r t i < 0 see 8 Substitutig 4 ito 7 we fid that h = P / + + j= i < <i k h p i p ik < /p j p j 3 + p j + p i p ik a i +r i+ p i p ik = P / + p i p ik a i +r h i+ p j + k+ p j 3 + p i p ik p i p ik + + o / l b i s i + + o / h 4, 5 where > s > > s l >. That is k= P / = l b i s i + + o /. 6 6

7 Note that all positive expoets i equatio 5, that is, the positive expoets of the form p j, 3 + p j, + r i + p j, p i p ik 3 +, + r i + p i p ik p i p ik are see 8 of the form m i i where m i ad i are relatively prime ad the i are squarefree itegers with prime divisors bouded by. Therefore these expoets are differet from ad cosequetly the expoets s i i =,...,l i 6 are of this same form. Note that + >, sice <. Cosequetly equatio 6 gives P = l b i s i + + o h /p = + + o + = 5/3 + l b i s i q a i r i + + c i k i + + o + h 4, 7 where > 5/3 > r > > r s > + > k > > k q > +. Note also that the first terms i equatio 7 are the terms of equatio 8. O the other had i equatio 7 appear the term + see equatio 8. We ow prove these facts. Equatio 7 ca be writte i the form P = Q + q c i k i + + o + = Q + o +, 8 where Q is a sum of terms of the form e i q i q i +. O the other had, equatio 8 ca be writte i the form P = 5/3 + a i r i + + o +. 9 Equatios 8 ad 9 give 0 = P P = Q 5/3 + a i r i + + o +. If Q 5/3 + a i r i + 7

8 the we obtai 0 = P P a q a 0 q +. That is, a evidet cotradictio. Cosequetly Q = 5/3 + a i r i +. 0 Fially, equatios 8 ad 0 give 7. Lemma 5 is costructive, we ca build the ext formula usig the former formula. Next, we build the formula that correspod to h = 4. We shall eed this formula. If h = 4 equatio 6 is see Nx = x / + x /3 + x /5 x /6 + + ox /7. O the other had Lemma 5 equatio 8 is see 5 Cosequetly equatio 5 is = P / + / Therefore + + o /7 = P / P = 5/3 + + o 7/5. = P / + /3 + /5 5 3 /3 + + o /7. Cosequetly see 7 + / / /3 3 /3 + /5 5 /5 / o/7 3 P / = /3 / /3 + + o /7. P = /3 / /3 + + o 9/7 That is = 5/3 7/ /6 + + o 9/7. P = 5/3 7/ /6 + + o 9/7. 8

9 4 Mai Result The followig theorem is the mai result of this article. I this theorem we obtai explicit formulae for P. Theorem 6. Let be the h-th prime with h 3, where h is a arbitrary but fixed positive iteger. Let us cosider the formula see We have h Nx = k+ k= i < <i k h p i p ik < P = /6 + 3 h 5 3/30 + k k= p x i p ik + + ox /. 3 i < <i k h p i p ik <, p i p ik, 6, 30 + p i p ik + + o +. 4 Proof. We shall see that everythig relies o Theorem. The theorem is true for h = 3 see Lemma 5 ad for h = 4 see ad. Suppose h 5, that is. Equatio 3 ca be writte i the form see Nx = x / + x /3 + x /5 x /6 + +a i x / i + + ox /, 5 where a i is the umber of differet prime factors i i ad the expoets are i decreasig order, > 3 > 5 > 6 > > > s >. 6 For example, if h = 5 the equatio 5 becomes equatio. O the other had, we have Lemma 5 ad equatio P = 5/3 7/ /6 + where the expoets are i decreasig order, d i r i + + o +, 7 > 5 3 > 7 5 > 8 6 > r > > r t > +. 8 Equatio 7 gives P = /3 3/ /6 + 9 d i r i + + o + 9

10 where /3 3/ /6 + d i r i + + o + /3, sice see 8 3 > 3 5 > 4 6 > r > > r t > Cosequetly Besides A = /3 3/ /6 + B = = d i r i + + o + = O /3 = o. 3 /3 3/ /6 + 3 /3 3/ o 4/6 d i r i + + o + = 4 / /5 + O. 3 Substitutig x = P ito equatio 5 ad usig Lemma 3 we obtai = P / + P /3 + P /5 P /6 + Equatios 9, 3, 3 ad Lemma give P / = + A + = /3 / /6 + Equatios 9, 3, 30 ad Lemma give +a i P / i + / + o /. 33 B + O d i r i / + o /p h /6 /30 + O. 34 P /3 = /3 + 3 A + O /3 = /3 3 /6 3 /30 + O. 35 P /5 = /5 + 5 A + O /3 = /5 5 /30 + o. 36 0

11 P / i = / i P /6 = /6 + 6 A + O /3 = /6 + O i A + O /3 = / i + o i =,...,s. 38 Substitutig equatios 34, 35, 36, 37 ad 38 ito equatio 33 we fid that 0 = d i r i + +a i / i 3 5 /30 + o /. 39 Note that see 8 ad 6 r i > ad i >. If 9 the 3 5 /30 = o /. Cosequetly we have d i r i = a i / i, where t = s, d i = a i i =,...,s ad r i = + i i =,...,s. Sice i cotrary case we have 0 a b where a 0 ad b >, a evidet cotradictio. Substitutig these values ito 7 we obtai 4 see 5. If 3 the > 30 ad there exists k such that k = 30 =.3.5 see 3. Cosequetly we have d i r i = a i / i /30, where t = s, d i = a i i k, d k = + 3 = 3 ad r 5 5 i = + i i =,...,s. Sice i cotrary case we have 0 a b where a 0 ad b >, a evidet cotradictio. Substitutig these values ito 7 we obtai 4 see 5. Example 7. If h = 5 equatio 3 is see Nx = x / + x /3 + x /5 x /6 + x /7 x /0 + + ox /. Cosequetly Theorem 6 gives P = 5/3 7/ /3 9/7 + 6/5 + + o 3/ 5 Ackowledgemets The author would like to thak the aoymous referee for his/her valuable commets ad suggestios for improvig the origial versio of this article. The author is also very grateful to Uiversidad Nacioal de Lujá.

12 Refereces [] E. Bach ad J. Soreso, Sieve algorithms for perfect power testig, Algorithmica 9 993, [] D. Berstei, Detectig perfect powers i essetially liear time, Math. Comp , [3] R. Jakimczuk, O the distributio of perfect powers, J. Iteger Seq. 4 0, Article.8.5. [4] P. Mihăilescu, A class umber free criterio for Catala s cojecture, J. Number Theory , 5 3. [5] P. Mihăilescu, Primary cyclotomic uits ad a proof of Catala s cojecture, J. Reie Agew. Math , [6] P. Mihăilescu, O the class groups of cyclotomic extesios i presece of a solutio to Catala s equatio, J. Number Theory 8 006, [7] M. A. Nyblom, A coutig fuctio for the sequece of perfect powers, Austral. Math. Soc. Gaz , [8] M. A. Nyblom, Coutig the perfect powers, Math. Spectrum 4 008, Mathematics Subject Classificatio: Primary A99; Secodary B99. Keywords: -th perfect power, asymptotic formula. Cocered with sequece A Received October 0; revised versios received Jauary 4 0; March 0 0; May 9 0. Published i Joural of Iteger Sequeces, May 9 0. Retur to Joural of Iteger Sequeces home page.