Coprime pairs with equal divisor sums from smooth successor primes

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smooth_successor_sign_audit: Every place where the sign of 2u+1 enters the weighted-dilation manuscript's proof of its smooth-predecessor count, with the identity, divisor bound, unit class, residue count or range that replaces it for primes 2u-1; the sign-free steps remain the manuscript's unrefereed argument.

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This folder's results that are not drawn from a cited source are Plasma AI's, credited to Plasma AI. The route itself is Shisheng Li and Zhong Zhuang's, recorded on their claim page: the reduction follows Pollack and Pomerance after an idea of Erdős, and the counting argument it adapts is OpenAI's weighted-dilation manuscript.

The complete conditional derivation writes out the signed Type II estimate, the sieve, and the squarefree counting reduction. It identifies the exact uniform analytic premise (P) and elementary source inputs (E), and includes an author examination of the analytic core. This page gives the route and the final counting argument; the linked derivation states the full endpoint interface.

Target

Problem 824, status open. Let $h(x)$ count the pairs $1\le a<b<x$ with $(a,b)=1$ and $\sigma(a)=\sigma(b)$. The question is whether $h(x)>x^{2-o(1)}$, that is, whether for every $\varepsilon>0$ one has $h(x)>x^{2-\varepsilon}$ for all large $x$. Pollack and Pomerance proved $h(x)>x^{1.4}$ (their Section 6 result). Nothing here answers the question unconditionally.

The conditional route

Write $P^+(m)$ for the largest prime factor of $m$. Hypothesis $\mathrm A^+$ is Definition 1.2 of Li and Zhuang's draft: for every $0<\delta<1/4$ and every $\eta>0$ there is $y_0$ such that every real $y\ge y_0$ has

$$\#\{p\text{ prime}:\ p\le5y,\ P^+(p+1)\le y^\delta\}\ge y^{1-\eta}.$$

The route has three links.

1. Premises. The weighted-dilation manuscript's lifted shift-cancellation Theorem 5.1 holds uniformly for the endpoints consumed by the [[research/smooth_successor_primes/signed_type_ii|signed Type II estimate]]; this is (P). Its Section 2 estimates, candidate mass Proposition 7.1 and proxy Lemmas 7.3–7.4 supply (E). Both remain unrefereed source premises. The [[research/smooth_successor_primes/shift_sign_analytic_core|author examination of the analytic core]] does not independently establish (P).
2. Sign change. Under these premises, for every fixed $0<\delta<1/4$ at least $x^{1-o(1)}$ primes $p$ with $2x-1\le p\le4x-1$ have $P^+(p+1)\le x^\delta$, and hence $\mathrm A^+$ holds. This is Li and Zhuang's Theorem 1.4, with a slightly narrower range. The sign audit lists every step of the manuscript's Sections 6 and 7 at which the sign enters and proves the replacement for primes $2u-1$: the determinant identity of the Type II estimate and its converse, the divisor-moment bounds, the unit class of the Type I estimate, the residue counts of the sieve for balanced composites, and the final range. This compact source comparison remains author-recorded; the full conditional derivation has the separate passing grade described below.
3. Reduction. $\mathrm A^+$ implies $h(x)>x^{2-\varepsilon}$ for every $\varepsilon>0$ and all large $x$, with $a,b$ squarefree. This is Li and Zhuang's Theorem 1.3, whose proof is rederived below.

Composed, the links give the requested bound conditional on (P) and (E). The conditional whole-conclusion grade matches the surviving E scopes to this chain. An unconditional answer still requires accepted P at its full consumed uniformity, together with the explicit classical boundaries. Li and Zhuang report a Lean proof of the implication in link 3, replayed by the Palomar registry; this corpus has not built or audited it, and it is not this corpus's Lean acceptance.

The reduction from Hypothesis A+

The argument is Li and Zhuang's Section 3, with every parameter checked here. Fix $0<\varepsilon<1$, a real $B\ge8/\varepsilon$, $\eta=\varepsilon/16$ and $\delta=1/(2B)\in(0,1/4)$. Write $L=\log x$, put $y=L^B$, let $\mathcal P$ be the set of primes $p\le5y$ with $P^+(p+1)\le y^\delta=L^{1/2}$, and $M=|\mathcal P|$. By $\mathrm A^+$, $M\ge L^{B(1-\eta)}$ for large $x$. Let $k=\lfloor L/\log(5y+1)\rfloor$, so that $(5y+1)^k\le x$ and $k=(1+o(1))L/(B\log L)$; also $M\ge2k$ for large $x$.

Let $\mathcal A$ be the set of products of $k$ distinct primes of $\mathcal P$. Each $a\in\mathcal A$ is squarefree and satisfies $a\le(5y)^k<(5y+1)^k\le x$. Distinct sets give distinct products, and

$$\log N:=\log|\mathcal A|\ge k\log(M/k) \ge L\Bigl(1-\eta-\frac1B\Bigr)(1+o(1)).$$

For $a=p_1\cdots p_k$, $\sigma(a)=\prod_i(p_i+1)\le x$ is $L^{1/2}$-smooth. An $L^{1/2}$-smooth integer up to $x$ is $\prod_{q\le L^{1/2}}q^{e_q}$ with $0\le e_q\le L/\log2$, so at most $U=(1+L/\log2)^{L^{1/2}}$ values occur and $\log U=o(L)$. With $r(s)=\#\{a\in\mathcal A:\sigma(a)=s\}$, Cauchy's inequality gives at least $N^2/U$ ordered pairs $(a,b)\in\mathcal A^2$ with $\sigma(a)=\sigma(b)$.

Let $t=\lceil\eta k\rceil$. Ordered pairs sharing at least $t$ primes number at most $N\binom kt\binom M{k-t}$, and since $M\ge2k$,

$$\binom M{k-t}\Big/\binom Mk=\prod_{i=0}^{t-1}\frac{k-i}{M-k+t-i} \le\Bigl(\frac{2k}M\Bigr)^t .$$

With $\binom kt\le2^k$, $k\le L$ and $M\ge L^{B(1-\eta)}$,

$$\log\Bigl(2^k\Bigl(\frac{2k}M\Bigr)^t\cdot2U\Bigr) \le(k+1)\log2+\log U-\eta k\bigl(B(1-\eta)\log L-\log(2L)\bigr) =-\eta L\Bigl(1-\eta-\frac1B\Bigr)(1+o(1))+o(L),$$

which is negative for large $x$. So at least $N^2/(2U)$ ordered pairs with $\sigma(a)=\sigma(b)$ share fewer than $t$ primes, and $a\ne b$ for them.

For such a pair put $g=(a,b)$, $u=a/g$, $v=b/g$. As $a,b$ are squarefree, $g$, $u$ and $v$ are pairwise coprime, so $\sigma(g)\sigma(u)=\sigma(g)\sigma(v)$ gives $\sigma(u)=\sigma(v)$, with $(u,v)=1$, $u\ne v$, and $u,v<x$ squarefree. Since $g$ is a product of at most $t-1<\eta k$ primes below $5y+1$, $g\le x^\eta$; each unordered pair $\{u,v\}$ therefore comes from at most $2x^\eta$ ordered pairs. Hence

$$h(x)\ge\frac{N^2}{4Ux^\eta},\qquad \log h(x)\ge\Bigl(2-3\eta-\frac2B\Bigr)L-o(L)>(2-\varepsilon)L$$

for large $x$, since $3\eta+2/B\le7\varepsilon/16$. Only the values of $\mathrm A^+$ at $\delta=1/(2B)$ and $\eta=\varepsilon/16$ are used, at the point $y=(\log x)^B$.

Standing and what remains

Links 2 and 3 have independently graded coverage through the full conditional chain. The whole conditional examination records refutation-failed for their composition from P and the operative E estimates, including every sufficiently large real input and the strict squarefree-pair conclusion; its distinct whole-contract grade is PASS, examined on 2026-10-16 UTC. The E examination has a separate passing grade for its adverse and surviving scopes, while the P grade remains VOID with zero accepted independent weight. The authors' externally replayed Lean proof covers the lower-bound implication for natural $x$, without the squarefree addendum. No build, axiom audit or statement-fidelity audit of it is supplied here. P remains without accepted independent weight; the accepted E scopes are relative to their quoted classical premises. The problem stays open. The route still depends on the unrefereed weighted-dilation manuscript; original classical proofs and PDF fidelity remain unexamined.

The combined sketch records the exact remaining uniformity: the discrepancy exponent must be chosen before the proxy's final cell precision, for all consumed endpoints, intervals, cutoffs and both shift signs. The two signs use a common permitted class, but their separated cutoff coefficients need not coincide. Establishing only selected predecessor instances does not justify the successor count. No sign-specific failure was found; that author observation supplies no independent acceptance of the analytic estimates.