A Note on Oppermann's Conjecture

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A Note on Oppermann's Conjecture

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Frank Vega^*

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Submitted:

18 August 2024

Posted:

21 August 2024

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Abstract

A prime gap is the difference between consecutive prime numbers. The $n^{\text{th}}$ prime gap, denoted $g_{n}$, is calculated by subtracting the $n^{\text{th}}$ prime from the $(n+1)^{\text{th}}$ prime: $g_{n}=p_{n+1}-p_{n}$. Oppermann's conjecture is a prominent unsolved problem in pure mathematics concerning prime gaps. Despite verification for numerous primes, a general proof remains elusive. If true, the conjecture implies that prime gaps grow at a rate bounded by $g_{n}<{\sqrt {p_{n}}}$. This note presents a proof of Oppermann's conjecture using the Euler-Maclaurin formula on harmonic numbers. This proof simultaneously establishes Andrica's, Legendre's, and Brocard's conjectures.

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

MSC: 11A41; 11A25

1. Introduction

Prime numbers, the fundamental building blocks of integers, have captivated mathematicians for centuries. Their erratic distribution, punctuated by seemingly random gaps, remains a captivating enigma. Several conjectures, including those related to large prime gaps, attempt to elucidate patterns within this irregularity by correlating prime gap sizes with the primes themselves.

Andrica’s conjecture, attributed to Dorin Andrica, posits a specific relationship between consecutive primes [1]. It asserts that the inequality

\sqrt{p_{n + 1}} - \sqrt{p_{n}} < 1

holds true for all positive integers n, where

p_{n}

represents the

n^{th}

prime number. Equivalently, if

g_{n}

denotes the

n^{th}

prime gap (the difference between

p_{n + 1}

and

p_{n}

), Andrica’s conjecture can be expressed as

g_{n} < 2 \cdot \sqrt{p_{n}} + 1 .

Legendre’s conjecture, attributed to Adrien-Marie Legendre, posits the existence of at least one prime number between the squares of any consecutive positive integers [2]. This unsolved problem is classified as one of Landau’s problems and implies that the gap between a prime and its successor is on the order of the square root of the prime (expressed as

O (\sqrt{p})

Oppermann’s conjecture, another open question related to prime distribution, is a stronger assertion than both Legendre’s and Andrica’s conjectures. Proposed by Danish mathematician Ludvig Oppermann in 1877, it suggests an upper bound for prime gaps of

g_{n} < \sqrt{p_{n}}

[3]. The conjecture states that, for every integer

x > 1

, there is at least one prime number between

x \cdot (x - 1) and x^{2},

and at least another prime between

x^{2} and x \cdot (x + 1) .

If true, this would also entail Brocard’s conjecture, which states that there are at least four primes between the squares of consecutive odd primes [2].

Despite its seemingly straightforward formulation, Oppermann’s conjecture has far-reaching implications for our comprehension of prime number distribution. Although extensively verified for countless primes, a general proof remains elusive. This unproven conjecture nonetheless serves as a compelling focal point, driving research to uncover deeper patterns in the prime number sequence. By resolving Oppermann’s conjecture, this work aims to significantly advance our understanding of this fundamental mathematical enigma.

2. Background and Ancillary Results

The Euler-Mascheroni constant, denoted by

γ \approx 0.57721

, is defined as the limit of the difference between the

n^{th}

harmonic number

H_{n}

and the natural logarithm of n as n approaches infinity [4]. The

n^{th}

harmonic number

H_{n}

is the sum of the reciprocals of the first n positive integers [4].

Proposition 1.

Building upon this constant, the Euler-Maclaurin formula provides an approximation for

H_{n}

involving γ,

log n

, and a rapidly decreasing error term

0 \leq ε_{n} \leq \frac{1}{8 \cdot n^{2}}

[5]:

H_{n} = log n + γ + \frac{1}{2 \cdot n} - ε_{n} .

Proposition 2.

Additionally, a logarithmic inequality holds for all positive values of t [6]:

\frac{1}{t + 0.5} < log (1 + \frac{1}{t}) .

By combining these results, we present a proof of Oppermann’s conjecture.

3. Main Result

This is a trivial result.

Lemma 1.

For every two consecutive primes

p_{n}

and

p_{n + 1}

, if the inequality

\sqrt{p_{n + 1}} - \sqrt{p_{n}} < \frac{1}{3}

holds then

g_{n} < \sqrt{p_{n}}

Proof.

The inequality

\sqrt{p_{n + 1}} - \sqrt{p_{n}} < \frac{1}{3}

is the same as

\sqrt{p_{n + 1}} < (\sqrt{p_{n}} + \frac{1}{3})

and

p_{n + 1} < {(\sqrt{p_{n}} + \frac{1}{3})}^{2}

after raising both sides to the square and distributing the terms. We know that

{(\sqrt{p_{n}} + \frac{1}{3})}^{2} = p_{n} + \frac{2}{3} \cdot \sqrt{p_{n}} + \frac{1}{9}

which is

g_{n} = p_{n + 1} - p_{n} < \frac{2}{3} \cdot \sqrt{p_{n}} + \frac{1}{9}

and so,

\frac{2}{3} \cdot \sqrt{p_{n}} + \frac{1}{9} < \sqrt{p_{n}}

for all

p_{n} \geq 2

. □

This is a key finding.

Lemma 2.

For every two consecutive primes

p_{n}

and

p_{n + 1}

, the following inequalities

H_{p_{n}} - log (p_{n}) \leq γ + \frac{1}{2 \cdot p_{n}}

log (p_{n + 1}) - H_{p_{n + 1}} \leq - γ - \frac{1}{2 \cdot p_{n + 1}} + \frac{1}{8 \cdot p_{n + 1}^{2}}

hold.

Proof.

By Proposition 1, we have

H_{p_{n}} = log p_{n} + γ + \frac{1}{2 \cdot p_{n}} - ε_{p_{n}}

for every prime

p_{n}

where

0 \leq ε_{p_{n}} \leq \frac{1}{8 \cdot p_{n}^{2}}

. Therefore, we have

H_{p_{n}} \leq log p_{n} + γ + \frac{1}{2 \cdot p_{n}}

and so,

H_{p_{n}} - log (p_{n}) \leq γ + \frac{1}{2 \cdot p_{n}} .

Again, using Proposition 1, we obtain

H_{p_{n + 1}} = log (p_{n + 1}) + γ + \frac{1}{2 \cdot p_{n + 1}} - ε_{p_{n + 1}}

for every prime

p_{n + 1} > 2

where

0 \leq ε_{p_{n + 1}} \leq \frac{1}{8 \cdot p_{n + 1}^{2}}

. So, we could show

H_{p_{n + 1}} \geq log (p_{n + 1}) + γ + \frac{1}{2 \cdot p_{n + 1}} - \frac{1}{8 \cdot p_{n + 1}^{2}}

and thus,

- γ - \frac{1}{2 \cdot p_{n + 1}} + \frac{1}{8 \cdot p_{n + 1}^{2}} \geq log (p_{n + 1}) - H_{p_{n + 1}} .

□

This is a main insight.

Lemma 3.

For every two consecutive primes

p_{n}

and

p_{n + 1}

, the inequality

\sqrt{p_{n + 1}} - \sqrt{p_{n}} < \frac{1}{3}

holds whenever

log (p_{n + 1}) - log (p_{n}) < \frac{2}{3 \cdot \sqrt{p_{n}} + 0.5}

holds as well.

Proof.

There is not any natural number

n^{'}

such that

\sqrt{p_{n^{'} + 1}} - \sqrt{p_{n^{'}}} = \frac{1}{3}

since this implies that

g_{n^{'}} = \frac{2}{3} \cdot \sqrt{p_{n^{'}}} + \frac{1}{9}

. For every n,

g_{n}

is a natural number and

\frac{2}{3} \cdot \sqrt{p_{n}} + \frac{1}{9}

is always irrational. In fact, all square roots of natural numbers, other than of perfect squares, are irrational [7]. Suppose that there exists a natural number

n_{0}

such that

\sqrt{p_{n_{0} + 1}} - \sqrt{p_{n_{0}}} > \frac{1}{3}

under the assumption that the inequality

log (p_{n_{0} + 1}) - log (p_{n_{0}}) < \frac{2}{3 \cdot \sqrt{p_{n_{0}}} + 0.5}

holds. That is equivalent to

\sqrt{\frac{p_{n_{0} + 1}}{p_{n_{0}}}} - 1 > \frac{1}{3 \cdot \sqrt{p_{n_{0}}}}

and

\sqrt{\frac{p_{n_{0} + 1}}{p_{n_{0}}}} > 1 + \frac{1}{3 \cdot \sqrt{p_{n_{0}}}}

after dividing both sides by

\sqrt{p_{n_{0}}}

and distributing the terms. We obtain that

log (p_{n_{0} + 1}) - log (p_{n_{0}}) > 2 \cdot log (1 + \frac{1}{3 \cdot \sqrt{p_{n_{0}}}})

if we apply the logarithm to the both sides. That would be the same as

log (p_{n_{0} + 1}) - log (p_{n_{0}}) > \frac{2}{3 \cdot \sqrt{p_{n_{0}}} + 0.5}

due to

log (1 + \frac{1}{3 \cdot \sqrt{p_{n_{0}}}}) > \frac{1}{3 \cdot \sqrt{p_{n_{0}}} + 0.5}

by Proposition 2. Since this implies that our initial assumption that

log (p_{n_{0} + 1}) - log (p_{n_{0}}) < \frac{2}{3 \cdot \sqrt{p_{n_{0}}} + 0.5}

should be false, we reach a contradiction. By reductio ad absurdum, we conclude that the Lemma 3 is true. □

This is the main theorem.

Theorem 1.

The Oppermann’s conjecture is true.

Proof.

We have confirmed the conjecture for

p_{n}

up to

10^{8}

by a numerical computation. Consequently, the Oppermann’s conjecture is true if the inequality

log (p_{n + 1}) - log (p_{n}) < \frac{2}{3 \cdot \sqrt{p_{n}} + 0.5}

holds for all

p_{n} > 10^{8}

as a direct consequence of Lemmas 1 and 3. For all

p_{n} > 10^{8}

, this inequality is equivalent to

log (p_{n + 1}) - H_{p_{n + 1}} + H_{p_{n}} - log (p_{n}) + \frac{1}{p_{n + 1}} < \frac{2}{3 \cdot \sqrt{p_{n}} + 0.5}

since

- H_{p_{n + 1}} + H_{p_{n}} + \frac{1}{p_{n + 1}} = 0

and therefore,

log (p_{n + 1}) - log (p_{n}) = log (p_{n + 1}) - H_{p_{n + 1}} + H_{p_{n}} - log (p_{n}) + \frac{1}{p_{n + 1}} .

By Lemma 2, we have

\begin{matrix} log (p_{n + 1}) - H_{p_{n + 1}} + H_{p_{n}} - log (p_{n}) + \frac{1}{p_{n + 1}} \\ \leq - γ - \frac{1}{2 \cdot p_{n + 1}} + \frac{1}{8 \cdot p_{n + 1}^{2}} + γ + \frac{1}{2 \cdot p_{n}} + \frac{1}{p_{n + 1}} \\ = - \frac{1}{2 \cdot p_{n + 1}} + \frac{1}{8 \cdot p_{n + 1}^{2}} + \frac{1}{2 \cdot p_{n}} + \frac{1}{p_{n + 1}} \\ = \frac{1}{2 \cdot p_{n + 1}} + \frac{1}{8 \cdot p_{n + 1}^{2}} + \frac{1}{2 \cdot p_{n}} . \end{matrix}

Hence, it is enough to show that

\frac{1}{2 \cdot p_{n + 1}} + \frac{1}{8 \cdot p_{n + 1}^{2}} + \frac{1}{2 \cdot p_{n}} < \frac{2}{3 \cdot \sqrt{p_{n}} + 0.5}

which is

0 < (\frac{1}{3 \cdot \sqrt{p_{n}} + 0.5} - \frac{1}{2 \cdot p_{n + 1}} - \frac{1}{8 \cdot p_{n + 1}^{2}}) + (\frac{1}{3 \cdot \sqrt{p_{n}} + 0.5} - \frac{1}{2 \cdot p_{n}})

that is trivially true because of

\begin{matrix} (\frac{1}{3 \cdot \sqrt{p_{n}} + 0.5} - \frac{1}{2 \cdot p_{n + 1}} - \frac{1}{8 \cdot p_{n + 1}^{2}}) & \geq (\frac{1}{3 \cdot \sqrt{p_{n}} + 0.5} - \frac{1}{2 \cdot (p_{n} + 2)} - \frac{1}{8 \cdot {(p_{n} + 2)}^{2}}) \\ > 0 \end{matrix}

and

(\frac{1}{3 \cdot \sqrt{p_{n}} + 0.5} - \frac{1}{2 \cdot p_{n}}) > 0

for all

p_{n} > 10^{8}

. □

4. Conclusions

This paper presents a novel approach to the longstanding Oppermann conjecture, leveraging the properties of harmonic numbers and the Euler-Maclaurin formula. By establishing a rigorous framework and employing careful analysis, we have demonstrated that the conjecture holds true for all prime numbers. This result not only resolves a fundamental open problem in number theory but also provides new insights into the distribution of primes. The implications of this work extend beyond prime number theory, potentially impacting areas such as cryptography, computational number theory, and related fields.

References

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A Note on Oppermann's Conjecture

Abstract

1. Introduction

2. Background and Ancillary Results

3. Main Result

4. Conclusions

References

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