The sequence of prime gaps is never strictly monotonic

Question

I have an assignment question that asks me to show that the sequence of prime gaps is never strictly monotonic. I'm also allowed to assume the Prime Number Theorem.

I've managed to show that it cannot be strictly decreasing by considering the numbers $N!+2, N!+3,...,N!+N$ which gets arbitrarily large as $N\rightarrow\infty$.

However, I seem to not be able to show the strictly increasing bit. I have an idea but I'm not sure if it works and whether it is a suitable usage of the Prime Number Theorem.

Here is my idea: Suppose $\pi(n_0)=k$ and $d(n)$ is strictly increasing $\forall n\geq n_0$. Then we consider the "worst case scenario" (call it $\pi_1$) of finding primes at gaps $2,4,6,...$ after $n_0$, meaning that $n_0+2,n_0+2+4,n_0+2+4+6,...$ are the primes.

Then what I'm saying is $\pi_1(n_0+q(q-1))=k+q-1$ because of the arithmetic progression. I don't know if we can give any comparisons between $\pi(n)$ and $\pi_1(n)$ but I also don't see if this actually leads to anything. If I'm totally off track, I'll be glad if somebody can point me in the right direction.

I can't seem to find any literature regarding this either!

Answer 1

As per your ides, there cannot be essentially more than $q$ primes below $q^2$ (give or take low order stuff like your constants $k$ and $n_0$), i.e. $\pi(q^2 )<q$. As $\pi(q^2)\approx \frac{q^2}{\ln q^2}=\frac q2\cdot \frac q{\ln q}\approx \frac q2\pi(q)$, this leads to $\pi(q)<2$, which is absurd. (Check that dropping low order terms was indeed allowed!)

Answer 2

It looks as if you're on a good track.

Suppose that after $p_n$, the gaps $d_n := p_{n+1} - p_n$ are strictly increasing, so that surely $d_{n+k} \geq k$. We can estimate $p_{n+k}$: $$p_{n+k} > d_{n+k-1} + \dots + d_{n} \geq (k-1) + (k-2) + \dots + 1 = \frac{k(k-1)}{2} \gg k^2.$$

But this is a problem, because $\pi(p_{n+k}) = n +k \ll k $ on one hand, and on the other $$\pi(p_{n+k}) \gg p_{n+k} / \log p_{n+k} \gg k^2/\log k.$$ It remains to convince yourself that these estimates are not compatible.

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Above $f(k) \ll g(k)$ stands for "there are constants $C$ and $k_0$ with $f(k) \leq C g(k)$ for $k \geq k_0$.

Note that the function $x/\log x$ is increasing, which is used above.