Composite Numbers with Large Prime Factors

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1 International Mathematical Forum, Vol. 4, 209, no., HIKARI Ltd, htts://doi.org/0.2988/imf Comosite Numbers with Large Prime Factors Rafael Jakimczuk División Matemática, Universidad Nacional de Luján Buenos Aires, Argentina Coyright c 209 Rafael Jakimczuk. This article is distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. Abstract Let k 3 an arbitrary but fied ositive integer. In this note we eamine the number of comosite numbers not eceeding such that all their rime factors are large, for eamle, greater than /k. Mathematics Subject Classification: A99, B99 Keywords: Numbers with large rime factors, distribution Preliminary Notes and Main Results We shall see that the numbers such that all their rime factors are large have zero density. That is, the number of these numbers not eceeding is o. Therefore, we are interested in to obtain more recise formulae. We need the following well-known Mertens s theorem. Lemma. The following asymtotic formula holds e γ + O log 2 Proof. See, for eamle, []. Let f be a ositive, continuous, strictly increasing function such that lim f

2 28 Rafael Jakimczuk and f. Now, let A be the number of ositive integers not eceeding such that all their rime factors are greater than f. We have the following theorem. Theorem.2 The following asymtotic formula holds. A e γ logf + O log 2 f Therefore A o. Proof. It is an immediate consequence of the inclusion-eclusion rincile and Lemma.. A + + E e γ logf f f + O log 2 f Since E The theorem is roved. f f f 2 log 2 f 2 f f If we ut f then we obtain the following corollary. Corollary.3 The following asymtotic formula holds. A e γ log + O log 2 Let π π be the number of rimes not eceeding. It is well-known the following equation rime number theorem π π + f where lim f 0. Let π m be the number of numbers not eceeding with eactly m rime factors in their rime factorization. It is well-known the following equation Landau s theorem π m log m m! 2

3 Comosite numbers with large rime factors 29 If m then equation 2 becomes equation. The following equation is well-known Mertens s theorem where M is Mertens s constant. log + M + o 3 Lemma.4 If 0 α β then the following limit holds. lim α β log log β α log β α 4 Proof. Let us consider the artition α α 0 α α 2 α n β, where c n α i+ α i β α n i 0,,..., n 5 Equation 3 gives α i α i+ log α i+ log α i + o i 0,,..., n 6 Now, the function has strictly decreasing derivative, consequently by the mean value theorem we have the inequality α i+ α i α i+ log α i+ log α i α i+ α i α i i 0,,..., n 7 Let us consider the inequality α i α i+ i 0,,..., n. 8 This inequality imlies the inequality α i log By integration theory we have α i+ i 0,,..., n 9 n lim n β α n α i+ α i lim α i α n i α i+ α i α i+ α i+ d A 0

4 30 Rafael Jakimczuk Let ɛ > 0. There eists n, sufficiently large and deending of ɛ, such that + c n β α i+ α i α i + c n α i + c n α i + c n α + ɛ ɛ + cn α α i α i+ + cn β 2 n α i+ α i A + b 3 α i α i where b ɛ n where a ɛ. Equations 9, 6 and 7 give α i+ α i+ α i+ α i A + a 4 α i+ α i + o α i α i+ α i α i+ log α i+ α i + o 5 α i+ α i Equations 5 and give α i α i+ log α i+ α i + o α i+ α i α i+ α i+ α i + o + ɛ α i+ α i α i+ α i+ α i α i+ α i+ + o 6 and equations 5 and 2 give α i α i+ log > ɛ α i+ α i + o 7 α i α i For sake of simlicity we ut A α β log 8

5 Comosite numbers with large rime factors 3 Therefore A α β log n α i α i+ log 9 From a certain value of we have see below o ɛ. Equations 9, 6 and 4 give n A + ɛ α i+ α i + o + ɛa + a α i+ α i+ + o + ɛa + ɛ + ɛ A + ɛa + + ɛɛ + A + 2ɛA + 20 Equations 9, 7 and 3 give n A ɛ α i+ α i + o ɛa + b α i α i + o ɛa ɛ ɛ A ɛa + ɛ ɛ A 2ɛA + 2 Therefore, since ɛ can be arbitrarily small, equations 8, 20 and 2 give equation 4. Note that A β α [ ] β β α d log log log α β α The theorem is roved. Lemma.5 Let us consider the comosite numbers not eceeding with eactly t 2 rime factors in their rime factorization such that each rime factor is greater than k and such that there is in the rime factorization some rime reeated, that is, some rime has multilicity greater than. The number of these comosite numbers not eceeding we denote λ t,k. The following formula holds. λ t,k o 22 Proof. If t 2 the roof is trivial since 2 imlies and consequently λ 2,k π log + o o

6 32 Rafael Jakimczuk If t 3 we can choose a reeated rime an therefore we have... s 2 where s t 2 and the i i,..., s are rime numbers. Consequently s Note that since the i i,..., s and are greater than k we have s k... s 2 k 24 and consequently 2 k s k s and s k log... s k 2 26 Therefore see 23, 24, 25, and 26 λ s,k π... s s k... s 2 k s k... s 2 k Ck Ck C... s log s k... s 2 k... s... s... s... s 27 If we ut f then we have see equation 2 and [] s... s log log t s 2 s s! t log t dt + o log s s! + o log s s! log log t s dt 2 s t log t 2 log s + o 28 s!

7 Comosite numbers with large rime factors 33 Since L Hosital s rule 2 lim log log t s 2 s t log t log s dt Equations 27 and 28 give equation 22. The theorem is roved. Now, we can rove our main theorems. Theorem.6 Let s + 2 an arbitrary but fied ositive integer. Let us consider the comosite numbers not eceeding with eactly s + different rime factors and such that each rime factor is greater than k. The number of these comosite numbers not eceeding we denote δ s+,k. The following formulae hold. If s + 2 then δ s+,k C s+,k 29 where C s+,k logk If s + 3 then there eist two ositive constants C and C 2 deending of s + and k such that C δ s+,k C 2 30 Proof. Note that k > s +. Let us consider a comosite number s s+ not eceeding with s + different rime factors i i,..., s + such that each rime factor is greater than k. That is, s s+ i > k i,..., s + 3 Equation 3 gives Note that k s+ s 32 s k s k 33 Equation 33 gives us and k s k 34 s s k log s k 35

8 34 Rafael Jakimczuk Equation 35 and equation 3 give k s s k k k k s s log s k s k s k s B 36 where B is a ositive constant. We can choose two finite sequences α, α 2,..., α s and β, β 2,..., β s such that k α β α 2 β 2 α s β s s k α + α α s β + β β s k Therefore equation 35 and equation 3 give k s s k s s k i s log s α i β i D s k s k s k s where D is a ositive constant. Equation 3, equation 32 and equation 33 give δ s+,k s + s k s k π π k Bs+,k 37 s where B s+,k is the number of comosite number with s + rime factors not eceeding, such that only a rime factor is reeated twice and such that each rime factor is greater than k. Therefore, by Lemma.5 we have B s+,k o 38 On the other hand, equation and equation 2 give π k s k s k π s k π k o 39

9 Comosite numbers with large rime factors 35 Equation gives + k s s k π s k s k s k s k s s log s f s s log s 40 Let ɛ > 0. There eists ɛ such that if ɛ see 34 we have f s ɛ. Therefore see 36 f s s log s ɛb s k s k and since ɛ can be arbitrarily small we have lim f s s log s 0 4 s k s k Equations 37, 38, 39, 40 and 4 give δ s+,k s + s k s k s log s + o 42 If s + 2 equation 42 and Lemma.4 give us equation 29. If s + 3 equation 42 and equations 36 give us equation 30. The theorem is roved. Theorem.7 Let k 3 an arbitrary but fied ositive integer. Let us consider the comosite numbers not eceeding such that each rime factor is greater than k. The number of these comosite numbers not eceeding we denote χ k. The following formulae hold. If k 3 then χ k log 2 43 If k 4 then the order of χ k is. That is, there eist two ositive constants D and D 2 deending of k such that D χ k D 2 44

10 36 Rafael Jakimczuk Proof. Note that the number of rime factors in these comosite numbers does not eceed k. Therefore the roof is an immediate consequence of Theorem.6 and Lemma.5. The theorem is roved. In the following theorem we rove a result stronger see equations 30 and 44. Theorem.8 The following asymtotic formulae hold. δ s+,k s +! L where L > 0 deends of s + and k and is defined bellow in the roof χ k D where D 3 > 0 can be easily eressed in terms of the L s constants. Proof. We shall rove equation 45, since equation 46 is an immediate consequence of equation 45. Let α s and β. We have see k k Theorem.6 and equation 42 A s k s k s log s Let us consider the artition α α 0 α α 2 α 2 n α i+ α i β α i 0,,..., 2 n. 2 n The inequality 47 β, where α i s α i+ 48 give us α i log s α i+ 49 Let us consider the following two functions deending of n. F 2 n, 2 n α i α i s α i+ s 50 F 2 2 n, 2 n α i+ α i s α i+ s 5

11 Comosite numbers with large rime factors 37 We have see equation 49 F 2 n, A F 2 2 n, 52 Now, A B see equation 36 therefore F 2 n, B. Note that 0 F 2 2 n, F 2 n, β α 2 n β 2 α s β s β α 2 n β M 53 2 since see equation 3 α s β s Equation 3 and integration theory give us α i s α i+ where the domain D i is k β s M 54 d d 2 d s + o 55 s s! D i 2 s α i s α i+, i k i, 2,..., s 56 Note that the frontier of D i has zero content and the set of oints, 2,, s in D i with reeated coordinates also has zero content. We also need, from the integration theory, a limit formula as 0 for functions of s variables. Note also that mean value theorem Π s i i Π s i + ilog i + i i Π s i i i i Equation 50 and equation 55 give F 2 n, s! 2 n α i D i 2 s d d 2 d s + o s! S n + o 57 where S n is the sequence S n 2 n α i D i 2 s d d 2 d s 58

12 38 Rafael Jakimczuk Note that since the function S n S n + 59 is strictly increasing in the interval 0, and d d 2 d s D i Di+ 2 s d d 2 d s + d d 2 d s 60 D i 2 s D i+ 2 s Since F 2 n, B see above equation 57 imlies that S n is bounded and hence there eists Equation 5 and equation 55 give F 2 2 n, s! 2 n lim S n L > 0 6 n α i+ D i 2 s d d 2 d s + o s! S 2n + o 62 where S 2 n is the sequence Note that S 2 n 2 n α i+ D i 2 s d d 2 d s 63 S 2 n > S 2 n + 64 Therefore there eists Equations 53, 57 and 62 give us Equations 52, 57 and 62 give lim S 2n L 2 65 n L L 2 L > 0 66 s! S n + o A s! S 2n + o 67 Let ɛ > 0. There eists n ɛ such that see equations 6, 65 and 66 S n ɛ L + a and S 2 n ɛ L + b, where a ɛ and b ɛ. On the other

13 Comosite numbers with large rime factors 39 hand from a certain value of we have o ɛ. Consequently equation 67 gives s! L 2ɛ s! L + a + o A s! L + b + o L + 2ɛ 68 s! Now, ɛ can be arbitrarily small. Therefore lim A s! L 69 Equations 42 and 69 give equation 45. The theorem is roved. Acknowledgements. The author is very grateful to Universidad Nacional de Luján. References [] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oford, 960. Received: January 5, 209; Published: January 28, 209