gpt pro 5.4 attempt on Math-5507

Problem 218 --- Round 2

1) ROUND-2 OBJECTIVE

Path (C): obstruction/correction. Round 1 isolated only elementary identities and the equivalence with consecutive primes in arithmetic progression, but did not produce any rigorous oscillation theorem for consecutive prime gaps. In this round I replace the still-open density-$1/2$ target by a theorem that is provable with current machinery: the strict comparison sets \[ A_{>} := \{n\ge 1 : d_{n+1}>d_n\}, \qquad A_{<} := \{n\ge 1 : d_{n+1}<d_n\} \] are both infinite, and in fact the ratio $d_{n+1}/d_n$ is unbounded above and has $0$ as a limit inferior.

2) Round-1 FOUNDATION USED

I use the Round 1 notation $d_n=p_{n+1}-p_n$ and the Round 1 density identity \[ \delta(\{d_{n+1}\ge d_n\})+\delta(\{d_{n+1}\le d_n\})=1+\delta(\{d_{n+1}=d_n\}) \] whenever the densities exist. I do not reprove that identity here.

3) NEW INSIGHT / TOOL (ROUND-2)

The new tool is Pintz's 2014 theorem on ratios of consecutive prime gaps. Specializing his Theorem 1 and Theorem 1$'$ to $k=1$ gives a quantitative form of oscillation: there is a constant $c>0$ such that for every sufficiently large $N$ there exists $n\in[N,2N]$ with \[ \frac{d_{n+1}}{d_n} > c\,\frac{\log N\,\log_2 N\,\log_4 N}{(\log_3 N)^2}, \] and also there exists $m\in[N,2N]$ with \[ \frac{d_{m-1}}{d_m} > c\,\frac{\log N\,\log_2 N\,\log_4 N}{(\log_3 N)^2}. \] Since the factor on the right tends to $\infty$, this is much stronger than merely saying that the sign of $d_{n+1}-d_n$ changes infinitely often.

4) ATTACK PLAN (ROUND-2)

The exact Round 1 gap was: there was no theorem showing that either strict inequality occurs infinitely often. To close that gap, it is enough to prove \[ \limsup_{n\to\infty}\frac{d_{n+1}}{d_n}=\infty, \qquad \liminf_{n\to\infty}\frac{d_{n+1}}{d_n}=0. \] These immediately imply $A_{>}$ and $A_{<}$ are both infinite. Pintz's theorem supplies exactly the needed dyadic-interval witnesses.

5) WORK (ROUND-2)

Let \[ L(N):=\frac{\log N\,\log_2 N\,\log_4 N}{(\log_3 N)^2}. \] By Pintz's Theorem 1 with $k=1$, there are constants $c>0$ and $N_0$ such that for every $N\ge N_0$ there exists $n\in[N,2N]$ with \[ \frac{d_{n+1}}{d_n}>cL(N). \] Now choose $N=2^j$ for $j$ sufficiently large. Then for each such $j$ there exists $n_j\in[2^j,2^{j+1}]$ with \[ \frac{d_{n_j+1}}{d_{n_j}}>cL(2^j). \] Because $L(2^j)\to\infty$, we obtain \[ \limsup_{n\to\infty}\frac{d_{n+1}}{d_n}=\infty. \] In particular $d_{n_j+1}>d_{n_j}$ for all large $j$, so $A_{>}$ is infinite.

Next apply Pintz's Theorem 1$'$ with $k=1$. There are (after possibly decreasing $c$ and enlarging $N_0$) indices $m_j\in[2^j,2^{j+1}]$ such that \[ \frac{d_{m_j-1}}{d_{m_j}}>cL(2^j). \] Set $r_j:=m_j-1$. Then $r_j\to\infty$ and \[ \frac{d_{r_j+1}}{d_{r_j}}<\frac{1}{cL(2^j)}\longrightarrow 0. \] Hence \[ \liminf_{n\to\infty}\frac{d_{n+1}}{d_n}=0. \] In particular $d_{r_j+1}<d_{r_j}$ for all large $j$, so $A_{<}$ is infinite.

Therefore we have proved the following corrected theorem.

Corrected theorem proved in Round 2. There exist infinitely many $n$ with $d_{n+1}>d_n$ and infinitely many $n$ with $d_{n+1}<d_n$; in fact, \[ \limsup_{n\to\infty}\frac{d_{n+1}}{d_n}=\infty, \qquad \liminf_{n\to\infty}\frac{d_{n+1}}{d_n}=0. \]

Combining this with the Round 1 density identity shows exactly where the original Erdős problem remains deeper: oscillation is known, but nothing here yields asymptotic frequencies, and in particular nothing here decides whether \[ \delta(\{d_{n+1}\ge d_n\})=\delta(\{d_{n+1}\le d_n\})=1/2. \]

6) ADVERSARIAL VERIFICATION

• Is $k=1$ allowed in Pintz's theorems? Yes: the statement is for an arbitrary fixed natural number $k$, so $k=1$ is admissible.
• Any index problem with $m_j-1$? No: for large $j$, $m_j\ge 2^j\ge 2$, so $r_j=m_j-1\ge 1$.
• Do we really get infinitely many distinct indices? Yes: choosing dyadic intervals $[2^j,2^{j+1}]$ forces $n_j$ and $m_j$ to escape to infinity.
• Could equality $d_{n+1}=d_n$ still occur infinitely often? Yes; the present argument says nothing about the set $E=\{n:d_{n+1}=d_n\}$. This is exactly one of the unresolved parts of the original problem.
• Does this imply any density statement? No. Sparse oscillation is much weaker than density-$1/2$.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED). The original density problem is still open, but Round 2 proves a genuine theorem absent from Round 1: \[ \liminf_{n\to\infty}\frac{d_{n+1}}{d_n}=0, \qquad \limsup_{n\to\infty}\frac{d_{n+1}}{d_n}=\infty, \] so both strict inequalities $d_{n+1}>d_n$ and $d_{n+1}<d_n$ occur infinitely often.

8) COMPLETION ESTIMATE

COMPLETION: 35%

9) REFERENCES

• J. Pintz, On the ratio of consecutive gaps between primes, arXiv:1406.2658.