mixedmath

Introduction

The primary purpose of this note is to collect a few hitherto unnoticed or unpublished results concerning gaps between powers of consecutive primes. The study of gaps between primes has attracted many mathematicians and led to many deep realizations in number theory. The literature is full of conjectures, both open and closed, concerning the nature of primes.

In a series of stunning developments, Zhang, Maynard, and Tao^{1 1}James Maynard. Small gaps between primes. Ann. of Math. (2), 181(1):383–413, 2015. ^{2 2} Yitang Zhang. Bounded gaps between primes. Ann. of Math. (2), 179(3):1121–1174, 2014. made the first major progress towards proving the prime $k$-tuple conjecture, and successfully proved the existence of infinitely many pairs of primes differing by a fixed number. As of now, the best known result is due to the massive collaborative Polymath8 project,^{3 3} D. H. J. Polymath. Variants of the {S}elberg sieve, and bounded intervals containing many primes. Res. Math. Sci., 1:Art. 12, 83, 2014. which showed that there are infinitely many pairs of primes of the form $p, p+246$. In the excellent expository article, ^{4 4} Andrew Granville. Primes in intervals of bounded length. Bull. Amer. Math. Soc. (N.S.), 52(2):171–222, 2015. Granville describes the history and ideas leading to this breakthrough, and also discusses some of the potential impact of the results. This note should be thought of as a few more results following from the ideas of Zhang, Maynard, Tao, and the Polymath8 project.

Throughout, $p_n$ will refer to the $n$th prime number. In a paper, ^{5 5} Dorin Andrica. Note on a conjecture in prime number theory. Studia Univ. Babe\c s-Bolyai Math., 31(4):44–48, 1986. Andrica conjectured that \begin{equation}\label{eq:Andrica_conj} \sqrt{p_{n+1}} - \sqrt{p_n} < 1 \end{equation} holds for all $n$. This conjecture, and related statements, is described in Guy's Unsolved Problems in Number Theory. ^{6 6} Richard K. Guy. Unsolved problems in number theory. Problem Books in Mathematics. Springer-Verlag, New York, third edition, 2004. It is quickly checked that this holds for primes up to $4.26 \cdot 10^{8}$ in sagemath

# Sage version 8.0.rc1
# started with `sage -ipython`

# sage has pari/GP, which can generate primes super quickly
from sage.all import primes_first_n

# import izip since we'll be zipping a huge list, and sage uses python2 which has
# non-iterable zip by default
from itertools import izip

# The magic number 23150000 appears because pari/GP can't compute
# primes above 436273290 due to fixed precision arithmetic
ps = primes_first_n(23150000)    # This is every prime up to 436006979

# Verify Andrica's Conjecture for all prime pairs = up to 436006979
gap = 0
for a,b in izip(ps[:-1], ps[1:]):
    if b**.5 - a**.5 > gap:
        A, B, gap = a, b, b**.5 - a**.5
        print(gap)
print("")
print(A)
print(B)

In approximately 20 seconds on my machine (so it would not be harder to go much higher, except that I would have to go beyond pari/GP to generate primes), this completes and prints out the following output.

0.317837245196
0.504017169931
0.670873479291

7
11

Thus the largest value of $\sqrt{p_{n+1}} - \sqrt{p_n}$ was merely $0.670\ldots$, and occurred on the gap between $7$ and $11$.

So it appears very likely that the conjecture is true. However it is also likely that new, novel ideas are necessary before the conjecture is decided.

Andrica's Conjecture can also be stated in terms of prime gaps. Let $g_n = p_{n+1} - p_n$ be the gap between the $n$th prime and the $(n+1)$st prime. Then Andrica's Conjecture is equivalent to the claim that $g_n < 2 \sqrt{p_n} + 1$. In this direction, the best known result is due to Baker, Harman, and Pintz, ^{7 7} R. C. Baker, G. Harman, and J. Pintz. The difference between consecutive primes. {II}. Proc. London Math. Soc. (3), 83(3):532–562, 2001. who show that $g_n \ll p_n^{0.525}$.

In 1985, Sandor ^{8 8} Joszsef Sandor. On certain sequences and series with applications in prime number theory. Gaz. Mat. Met. Inf, 6:1–2, 1985. proved that \begin{equation}\label{eq:Sandor} \liminf_{n \to \infty} \sqrt[4]{p_n} (\sqrt{p_{n+1}} - \sqrt{p_n}) = 0. \end{equation} The close relation to Andrica's Conjecture \eqref{eq:Andrica_conj} is clear. The first result of this note is to strengthen this result.

Theorem Let $\alpha, \beta \geq 0$, and $\alpha + \beta < 1$. Then \begin{equation}\label{eq:main} \liminf_{n \to \infty} p_n^\beta (p_{n+1}^\alpha - p_n^\alpha) = 0. \end{equation}

We prove this theorem below. Choosing $\alpha = \frac{1}{2}, \beta = \frac{1}{4}$ verifies Sandor's result \eqref{eq:Sandor}. But choosing $\alpha = \frac{1}{2}, \beta = \frac{1}{2} - \epsilon$ for a small $\epsilon > 0$ gives stronger results.

This theorem leads naturally to the following conjecture.

Conjecture For any $0 \leq \alpha < 1$, there exists a constant $C(\alpha)$ such that \begin{equation} p_{n+1}^\alpha - p_{n}^\alpha \leq C(\alpha) \end{equation} for all $n$.

A simple heuristic argument, given in the last section below, shows that this Conjecture follows from Cramer's Conjecture.

It is interesting to note that there are generalizations of Andrica's Conjecture. One can ask what the smallest $\gamma$ is such that \begin{equation} p_{n+1}^{\gamma} - p_n^{\gamma} = 1 \end{equation} has a solution. This is known as the Smarandache Conjecture, and it is believed that the smallest such $\gamma$ is approximately \begin{equation} \gamma \approx 0.5671481302539\ldots \end{equation} The digits of this constant, sometimes called 'the Smarandache constant,' are the contents of sequence A038458 on the OEIS. It is possible to generalize this question as well.

Open Question For any fixed constant $C$, what is the smallest $\alpha = \alpha(C)$ such that \begin{equation} p_{n+1}^\alpha - p_n^\alpha = C \end{equation} has solutions? In particular, how does $\alpha(C)$ behave as a function of $C$?

This question does not seem to have been approached in any sort of generality, aside from the case when $C = 1$.

Proof of Theorem

The idea of the proof is very straightforward. We estimate \eqref{eq:main} across prime pairs $p, p+246$, relying on the recent proof from Polymath8 that infinitely many such primes exist.

Fix $\alpha, \beta \geq 0$ with $\alpha + \beta < 1$. Applying the mean value theorem of calculus on the function $x \mapsto x^\alpha$ shows that \begin{align} p^\beta \big( (p+246)^\alpha - p^\alpha \big) &= p^\beta \cdot 246 \alpha q^{\alpha - 1} \\\ &\leq p^\beta \cdot 246 \alpha p^{\alpha - 1} = 246 \alpha p^{\alpha + \beta - 1}, \label{eq:bound} \end{align} for some $q \in [p, p+246]$. Passing to the inequality in the second line is done by realizing that $q^{\alpha - 1}$ is a decreasing function in $q$. As $\alpha + \beta - 1 < 0$, as $p \to \infty$ we see that\eqref{eq:bound} goes to zero.

Therefore \begin{equation} \liminf_{n \to \infty} p_n^\beta (p_{n+1}^\alpha - p_n^\alpha) = 0, \end{equation} as was to be proved.

Further Heuristics

Cramer's Conjecture states that there exists a constant $C$ such that for all sufficiently large $n$, \begin{equation} p_{n+1} - p_n < C(\log n)^2. \end{equation} Thus for a sufficiently large prime $p$, the subsequent prime is at most $p + C (\log p)^2$. Performing a similar estimation as above shows that \begin{equation} (p + C (\log p)^2)^\alpha - p^\alpha \leq C (\log p)^2 \alpha p^{\alpha - 1} = C \alpha \frac{(\log p)^2}{p^{1 - \alpha}}. \end{equation} As the right hand side vanishes as $p \to \infty$, we see that it is natural to expect that the main Conjecture above is true. More generally, we should expect the following, stronger conjecture.

Conjecture' For any $\alpha, \beta \geq 0$ with $\alpha + \beta < 1$, there exists a constant $C(\alpha, \beta)$ such that \begin{equation} p_n^\beta (p_{n+1}^\alpha - p_n^\alpha) \leq C(\alpha, \beta). \end{equation}

Additional Notes

I wrote this note in between waiting in never-ending queues while I sort out my internet service and other mundane activities necessary upon moving to another country. I had just read some papers on the arXiv, and I noticed a paper which referred to unknown statuses concerning Andrica's Conjecture. So then I sat down and wrote this up.

I am somewhat interested in qualitative information concerning the Open Question in the introduction, and I may return to this subject unless someone beats me to it.

This note is (mostly, minus the code) available as a pdf and (will shortly) appears on the arXiv. This was originally written in LaTeX and converted for display on this site using a  set of tools I've written based around latex2jax, which is available on my github.

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