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volume 4, issue 1, article 23, 2003.

Received 10 January, 2003;

accepted 19 February, 2003.

Communicated by:J. Sándor

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Journal of Inequalities in Pure and Applied Mathematics

ERD ˝OS-TURÁN TYPE INEQUALITIES

LAUREN ¸TIU PANAITOPOL

University of Bucharest Faculty of Mathematics 14 Academiei St.

RO–70109 Bucharest Romania.

EMail:pan@al.math.unibuc.ro

c

2000Victoria University ISSN (electronic): 1443-5756 004-03

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Erd ˝os-Turán Type Inequalities Lauren¸tiu Panaitopol

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J. Ineq. Pure and Appl. Math. 4(1) Art. 23, 2003

http://jipam.vu.edu.au

Abstract

Denoting by(r_n)n≥1the increasing sequence of the numbersp^αwithpprime andα≥2integer, we prove thatrn+1−2rn+rn−1is positive for infinitely many values ofnand negative also for infinitely many values ofn. We prove similar properties forr²_n−rn−1r_n+1and_r¹

n−1−_r²

n+_r¹

n+1 as well.

2000 Mathematics Subject Classification:11A25, 11N05.

Key words: Powers of prime numbers, Inequalities, Erd ˝os-Turán theorems.

1. Introduction

Let(r_n)n≥0be the increasing sequence of the powers of prime numbers (p^αwith pprime and α ≥ 2integer). Thus, we haver1 = 4, r2 = 8, r3 = 9, r4 = 16, etc. Properties of the sequence(r_n)n≥1 were studied in [5] and [3].

Denote by p_n the n-th prime number. In [1], Erd˝os and Turán proved that p_n+1−2p_n+pn−1 is positive for infinitely many values ofn and negative also for infinitely many values ofn. Until now, no answer is known for the following question raised by Erd˝os and Turán: Do there exist infinitely many numbers n such that

p_n+1−2p_n+pn−1 = 0?

Erd˝os and Turán also proved that each of the sequences(p²_n−pn−1p_n+1)n≥2 and 1

pn−1 − _p²

n + _p¹

n+1

n≥2 has infinitely many positive terms and infinitely many negative ones.

Denoting by (q_n)n≥1 the increasing sequence of the powers of prime numbers, the author proved in [4] that the value of q_n+1 −2q_n +qn−1 changes its sign infinitely many times.

In the present paper, we raise similar problems for the sequence(r_n)n≥1. We need a few preliminary properties, which will be proved in the next section.

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2. On the Difference r

_n+1

− r

_n

Property 2.1. We have

(2.1) lim sup

n→∞

(r_n+1−r_n) =∞.

Proof. Letm≥4. We show that, among the numbers m! + 2, m! + 3, . . . , m! + [√

m], there is no term of the sequence(r_n)_n≥1.

Assume that there exists an integerasuch that2≤a ≤[√

m]and

(2.2) m! +a =pⁱ

wherepis prime andi≥2.

The relation (2.2) can also be written in the form a

m!

a + 1

=pⁱ, whencea =p^j with1≤j ≤i.

It follows that m!

p^j + 1 =p^i−j, hence m!

p^j is not divisible byp.

Ife_p(n)is Legendre’s function, we havee_p(m) = j, that is,

(2.3)

∞

X

s=1

m p^s

=j.

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Since a ≤ √

m, it follows that p^j ≤ √

m, that is, m ≥ p^2j, and then (2.3) implies that

j ≥ p^2j

p

+ p^2j

p²

+· · ·+ p^2j

p^2j

=p^2j−1+p^2j−2+· · ·+p+ 1

≥2^2j−1+ 2^2j−2+· · ·+ 2 + 1

= 2^2j−1.

Since forj ≥1we have2^2j−1> j, we obtained a contradiction.

Since our assumption turned out to be false, it follows that for everym ≥ 4 there existsk =k(m)such that

r_k ≤m! + 1andr_k+1 ≥m! + [√

m] + 1, whencer_k+1−r_k ≥[√

m], and finally lim sup

n→∞

(r_n+1−r_n) =∞,

and the proof ends.

We now denotea_n = ^r_nⁿ⁺¹_log^−r2nⁿ and recall that, in [2], H. Meier proved that

(2.4) lim inf

n→∞

p_n+1−p_n

logn <0.248.

In connection with this result, we prove:

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Property 2.2. We have

(2.5) lim inf

n→∞ a_n<0.496.

Proof. We consider the indicesmsuch that p_m+1−p_m

logm <0.248.

Both the numbersp²_mandp²_m+1occur in the sequence(rn)n≥1, that is,p²_m =rk

andp²_m+1 =r_h, withk =k(m),h=h(m)andh≥k+ 1. In [5], it was proved that, form≥1783, we have

(2.6) p²_m ≥r_m > m²log²m.

Since pm ∼ mlogm, it follows that rk ∼ k²log²k. But rk = p²_m, hence klogk ∼ mlogm. One can show without difficulty that k(m) ∼ m. It then follows that √

rk+1−√ rk

logk <

√r_h−√ r_k

logk = p_m+1−p_m logk . Sincelogk ∼logm, we get

lim inf

k→∞

√r_k+1−√ r_k

logk ≤lim inf

m→∞

pm+1−pm

logm <0.248.

Since√

r_k∼klogkand√

r_k+1 ∼(k+ 1) log(k+ 1)∼klogk, it follows that

lim inf

k→∞

r_k+1−r_k

klog²k <0.496,

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wherek=k(m). Consequently,

lim inf

n→∞

r_n+1−r_n

nlog²n <0.496.

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3. Erd˝os-Turán Type Properties

Fork≥2we denote

Rk=rk+1−2rk+rk−1, and prove

Property 3.1. There exist infinitely many values ofnsuch that R_n>0,

and also infinitely many ones such that

R_n<0.

Proof. DenotingS_m =Pm

k=2R_k, we haveS_m =r_m+1−r_m−r₂+ 1. By (2.1) we havelim sup

m→∞

S_m =∞, henceR_n >0for infinitely many values ofn.

Denotingσm =Pm

k=2kRk, we have

σ_m =m(r_m+1−r_m)−r_m−r₂+ 2r₁ =m²log²m

a_m− r_m m²log²m

.

Sincer_m ∼ m²log²m, we get by (2.5) that lim inf

m→∞ σ_m = −∞, henceR_n < 0 for infinitely many values ofn.

Fork ≥2, denotingρ_k= _r¹

k−1 − _r²

k + _r¹

k+1, we have

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Property 3.2. There exist infinitely many values ofnsuch that ρ_n>0,

and also infinitely many ones such that ρ_n<0.

Proof. Forα >3, denotingS_m⁰ (α) =

m

P

k=2

k^αρk, we get

S_m⁰ (α) = −m^α(r_m+1−r_m)

r_mr_m+1 − m^α−(m−1)^α r_m +

m−1

X

k=2

k^α−2(k−1)^α+ (k−2)^α

r_k +O(1).

We have

r_k ∼k²log²k, k^α−(k−1)^α ∼αk^α−1,

k^α−2(k−1)^α+ (k−2)^α ∼α(α−1)k^α−2, whence

m^α(r_m+1−r_m)

r_mr_m+1 ∼ m^α−3a_m log²m , m^α−(m−1)^α

rm

∼ αm^α−3 log²m, k^α−2(k−1)^α+ (k−2)^α

r_k ∼ α(α−1)k^α−4 log²k .

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Since

m−1

X

k=2

k^α−4

log²m ∼ (α−3)m^α−3 log²m , it follows that

S_m⁰ (α)∼ m^α−3

log²m · −a_m−α+α(α−1)(α−3) . Then lim

m→∞S_m⁰ (3.1) = −∞, and thus there exist infinitely many values of n such thatρ_n <0.

On the other hand, we have by (2.5) thatlim sup

m→∞

S_m⁰ (4) =∞, which shows that there exist infinitely many values ofnsuch thatρn>0.

A consequence of Properties3.1and3.2is the following.

Property 3.3. There exist infinitely many values ofnsuch that rn−1r_n+1 > r²_n,

and also infinitely many ones such that

rn−1r_n+1 < r²_n. Proof. If r_n > ^rⁿ⁺¹^+r₂ ⁿ⁻¹, then r_n > √

rn−1r_n+1. On the other hand, if _r²

n >

1

rn−1 + _r¹

n+1, then

rn <2

1 rn−1

+ 1 r_n+1

<√

rn−1rn+1, and then the desired conclusion follows by Properties3.1and3.2.

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Open problem. Do there exist infinitely many values ofnsuch that r_n+1−2r_n+r_n−1 = 0?

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References

[1] P. ERD ˝OS AND P. TURÁN, On some new question on the distribution of prime numbers, Bull. Amer. Math. Soc., 54(4) (1948), 371–378.

[2] H. MEIER, Small difference between prime numbers, Michigan Math. J., 35 (1988), 324–344.

[3] G. MINCU, An asymptotic expansion, (in press).

[4] L. PANAITOPOL, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain J. Math., 31(4) (2001), 1407–1415.

[5] L. PANAITOPOL, The sequence of the powers of prime numbers revisited, Math. Reports, 5(55)(1) (2003), (in press).