gpt pro 5.4 attempt on Math-784

Erdős Problem 371

Problem statement.

Let $P^{+}(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P^{+}(n)<P^{+}(n+1)$ has density $1/2$.

Tools in this round.

Web browsing: YES. Computation: YES.

1) ROUND-2 OBJECTIVE

I am not pursuing path (A), because the full natural-density statement is still open. The Round-2 goal is therefore strict advancement: replace Round 1's mere infinitude results by bona fide density theorems, and isolate the exact remaining obstruction between what is known and the desired limit $1/2$.

2) Round-1 FOUNDATION USED

I rely on two Round-1 facts.

• For every $n\geq 1$, one has $P^{+}(n)\neq P^{+}(n+1)$, since consecutive integers are coprime.
• Round 1 gave infinite families in both directions: $P^{+}(2^{m})<P^{+}(2^{m}+1)$ and $P^{+}(2^{m}-1)>P^{+}(2^{m})$.

These facts will now be strengthened to genuine density statements.

3) NEW INSIGHT / TOOL (ROUND-2)

The new ingredients are three published theorems.

• Ter"av"ainen's window theorem: for any exponents $0<a<b<1$ and $0<c<d<1$, the set of $n$ for which $P^{+}(n)$ and $P^{+}(n+1)$ lie simultaneously in the windows $[n^{a},n^{b}]$ and $[n^{c},n^{d}]$ has positive asymptotic lower density.
• Ter"av"ainen's logarithmic-density theorem: the set $\{n:P^{+}(n)<P^{+}(n+1)\}$ has logarithmic density $1/2$.
• Tao--Ter"av"ainen's almost-all-scales theorem, together with the explicit lower-density bound $0.2017$ of L"u--Wang.

The first item lets me prove positive lower density for both inequalities directly, which is already a substantial advance over Round 1.

4) ATTACK PLAN (ROUND-2)

The exact gap left after Round 1 was the absence of any mean-cancellation theorem for the sign sequence \[ f(n)=\begin{cases} +1,& P^{+}(n)<P^{+}(n+1),\\ -1,& P^{+}(n)>P^{+}(n+1). \end{cases} \] To prove density $1/2$ one would need \[ \sum_{n\leq X} f(n)=o(X). \] I do not have that. Instead I will do the next best rigorous thing.

1. Use Ter"av"ainen's window theorem to obtain positive lower density for both sides.
2. Record the stronger known theorems: logarithmic density $1/2$, density $1/2$ at almost all scales, and explicit lower density at least $0.2017$.
3. Explain precisely why these do not yet imply the full natural density statement.

5) WORK (ROUND-2)

Let \[ S=\{n\geq 1: P^{+}(n)<P^{+}(n+1)\}. \] Write \[ \underline d(A)=\liminf_{X\to\infty} \frac{1}{X}\#(A\cap [1,X]) \] for asymptotic lower density.

Step 1: a fully proved new theorem -- both sides have positive lower density.

Ter"av"ainen proved the following.

If $a,b,c,d\in (0,1)$ with $a<b$ and $c<d$, then the set \[ E_{a,b;c,d}=\{n\geq 1: n^{a}\leq P^{+}(n)\leq n^{b},\; n^{c}\leq P^{+}(n+1)\leq n^{d}\} \] has positive asymptotic lower density.

Choose, for example, \[ a=\frac{1}{10},\qquad b=\frac{1}{5},\qquad c=\frac{4}{5},\qquad d=\frac{9}{10}. \] Then $a<b$ and $c<d$, so $E_{a,b;c,d}$ has positive lower density. For every $n\in E_{a,b;c,d}$ we have \[ P^{+}(n)\leq n^{1/5}<n^{4/5}\leq P^{+}(n+1), \] whence $n\in S$. Therefore \[ \underline d(S)>0. \]

Now reverse the windows. Let \[ F=E_{4/5,9/10;1/10,1/5}. \] Again Ter"av"ainen's theorem gives $\underline d(F)>0$, and for every $n\in F$ one has \[ P^{+}(n)\geq n^{4/5}>n^{1/5}\geq P^{+}(n+1), \] so $n\notin S$. Hence \[ \underline d(\mathbf N\setminus S)>0. \] This strictly strengthens the Round-1 infinitude result.

Step 2: logarithmic density is exactly $1/2$.

Ter"av"ainen also proved that the logarithmic density \[ \delta_{\log}(A)=\lim_{X\to\infty}\frac{1}{\log X}\sum_{\substack{n\leq X\\ n\in A}}\frac{1}{n} \] exists for $A=S$ and satisfies \[ \delta_{\log}(S)=\frac{1}{2}. \] So the conjectured symmetry is already known on logarithmic average.

Step 3: at almost all scales, the ordinary density is also $1/2$.

Tao and Ter"av"ainen proved that there exists an exceptional set $\mathcal X_{0}$ of logarithmic density $0$ such that \[ \lim_{\substack{X\to\infty\\ X\notin \mathcal X_{0}}} \frac{1}{X}\#\{n\leq X: n\in S\}=\frac{1}{2}. \] Thus the natural-density statement is true for all scales outside a logarithmically negligible exceptional set.

Step 4: the best currently published explicit lower bound.

L"u and Wang proved \[ \#\{n<x:P^{+}(n)<P^{+}(n+1)\}>(0.2017-o(1))x, \] and the same lower bound for the complementary inequality. Therefore \[ \underline d(S)\geq 0.2017, \qquad \underline d(\mathbf N\setminus S)\geq 0.2017. \] Since $S$ and its complement partition the positive integers by the Round-1 no-ties observation, one also gets the corresponding upper bounds \[ \overline d(S)\leq 0.7983, \qquad \overline d(\mathbf N\setminus S)\leq 0.7983. \]

6) ADVERSARIAL VERIFICATION

The points most likely to conceal a mistake are the following.

• Window inclusion. In Step 1 the inequalities are strict for $n>1$ because $1/5<4/5$. So every $n$ in the first window set really belongs to $S$, and every $n$ in the reversed window set really belongs to the complement.
• Density notions. Positive lower density, logarithmic density, and "density at almost all scales" are three different notions. None of Steps 1--4 silently identifies them.
• Why don't Steps 2 and 3 already prove the conjecture? Because a set of scales of logarithmic density $0$ can still contain subsequences along which the ordinary Cesàro averages oscillate. So the Tao--Ter"av"ainen theorem is stronger than anything in Round 1, but weaker than the existence of an ordinary limit along all $X$.
• Compatibility with Round 1. The new results strengthen, rather than contradict, Round 1. Infinitude follows immediately from positive lower density.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED)

Round 2 upgrades the Round-1 output substantially:

• both inequalities now have rigorously proved positive lower density;
• the set $S$ is known to have logarithmic density $1/2$;
• the ordinary density equals $1/2$ at almost all scales;
• the best explicit published lower bound is $0.2017$ for both $S$ and its complement.

The exact missing step is now sharply isolated: one must upgrade these partial density results to an ordinary asymptotic density along all scales.

8) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 70%

9) REFERENCES

• J. Ter"av"ainen, On binary correlations of multiplicative functions, Forum Math. Sigma 6 $2018$, e10.
• T. Tao and J. Ter"av"ainen, The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures, Algebra & Number Theory 13 $2019$, no. 9.
• X. L"u and Z. Wang, On the largest prime factors of consecutive integers, Monatsh. Math. 206 $2025$, 403--418.
• T. F. Bloom, Erdős Problem #371, accessed 2026-03-07.

10) OUTPUT FORMAT

Erdős problem number: $x=371$. This section is written as a drop-in LaTeX fragment and is included in the combined file 369_371_373_round2_solutions.tex.

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