RESULTS: Erdős #126, literature arm (`support_fcffa3b9`)

Support run for 0897a5a7 / r_186989. A literature map, not a new theorem. Nothing here is evidence for or against RH (docs/08).

Notation throughout, following the parent hunt: $S$ is a set of $k$ primes, $g(k)$ is the largest $|A|$ over sets $A$ of distinct positive integers all of whose off-diagonal sums $a+b$ are $S$-smooth, maximised over $S$. Erdős #126 asks whether $f(n)/\log n \to \infty$, equivalently $g(k) = \exp(o(k))$.

Landing note (2026-10-21). Landed on main as Hunt #143 from a local-only branch, support/fcffa3b9, that was never pushed; PR #144 counted this arm among the seven that "produced no branch at all". The run's brief assigned #91, which r_186989 already held, so the run took #92 on its branch; #92 is support_517b887f on main, so the entry takes the next free number. HANDBACK.json keeps the run's own record of #92. The run's files are as it wrote them apart from punctuation and the dated notes marked as added on landing.

Grading (added on landing, 2026-10-21). This is a literature map from a targeted search (the problem page's bibliography, a forward and backward citation chase from GST 1986, the $S$-unit counting literature, an OEIS sequence query and keyword search), not an exhaustive one, and it closes no route. Primary text was read for T1, T1′ and T1″ (Erdős–Turán 1934) and for T3 (Győry–Stewart–Tijdeman 1986, including the §3 proof). T2 (Erdős–Surányi's $2^k$) and T5 (the Erdős–Stewart–Tijdeman constant $1/8$, and with it the $4+o(1)$ of §0 and §2(d)) are secondary citations through Füredi–Gyarmati 2026 and were not checked against the originals. T4 (Evertse's $3\cdot7^{2w+3}$) is primary only as quoted inside GST; T6, T7 and T9 are secondary. Step (a) of §2 is this lab's ordinary argument, unreviewed and pending external verification; step (b) is GST's, quoted. Nothing here is kernel-checked, and no test in tests/ pins anything in this directory.

Overlap with arms already on main. This arm ran in the same fan-out as Hunts #92 to #107, and most of its mathematics reached main by other routes. The two-set reduction of §2 is the reduction of Hunts #96 (support_d5d5ccae) and #104 (support_f3ab3e34), and of three sibling arms landed with this one (Hunts #97, #99 and #106). §4.2's 3/2-corridor point about the composition law is Hunt #105's (support_eccd5f5e), and Hunt #99 (support_311c14a5 §5, landed with this one) sharpens it to $g(k)=2^k$ exactly. #105 already cites GST 1986, EST 1988 and Füredi–Gyarmati. The frozen constant this arm ranks as its door ($|B|=2$) is Hunt #104's first door (its $m$-base-point version), while Hunt #96 argues that fixing three or more base elements is strictly worse under current counting bounds, because the extra coefficients carry primes outside $S$. What this arm adds is the primary-text reading: GST's reduction quoted from its own proof, the 1934 authors' conjecture $n(k)=O(k^{1+\varepsilon})$ (T1″), the parity argument on page 1 of the 1934 paper (§4.1), and the negative OEIS check (§4.4) that closes the OEIS thread Hunts #99 and #106 left open.

0. The answer, in five lines

1. The 1934 bound is still the record. $g(k) \le 3\cdot 2^{k-1} - 1$ (Erdős–Turán, primary text read). Nothing found in 92 years improves the base 2; the only improvement located is the constant, to $g(k) \le 2^k$ (Erdős–Surányi's book, secondary citation).
2. The $S$-unit machinery is in the literature and it is quantitatively worse, not better: Győry–Stewart–Tijdeman 1986 reproves $f(n) \gg \log n$ through Evertse's bound and lands at $g(k) \le 3\cdot 7^{2k+3} \approx 49^k$.
3. It is nevertheless the right lever, because the whole loss is in one named quantity: the number of solutions of a two-variable $S$-unit equation. A uniform bound $\exp(o(|S|))$ on that count implies Erdős #126. That is the closest theorem, and it is a known open problem in Diophantine approximation, not a lemma.
4. That route is not blocked. Erdős–Stewart–Tijdeman 1988 proves the relaxed two-set problem has $\exp\big((4+o(1))\sqrt{k/\log k}\big)$-sized witnesses, a hard obstruction to polynomial bounds, and still comfortably inside $\exp(o(k))$. The entire gap between $\exp(\tilde\Theta(\sqrt k))$ and $\exp(O(k))$ is open, and any part of it settles #126.
5. What Erdős and Turán actually conjectured in 1934 is far stronger than #126: $n(k) = O(k^{1+\varepsilon})$, i.e. $f(n) = n^{1-o(1)}$. #126 is the first inch of that.

1. Citation-checked map

Verification status is stated per row. "primary" = I read the paper text. "secondary" = stated in a paper I read, but I did not read the original.

The problem page (erdosproblems.com/126, accessed 2026-08-24, T. F. Bloom): open, $250 prize, cites only [ErTu34] [Er95c] [Er97] [Er97e], and records $\log n \ll f(n) \ll n/\log n$ with no intermediate result. A February 2026 paper on the problem (T10) writes the same history and cites no improvement of T1 or T3 either. That is two independent recent surveys of the same wall.

No literature result found improves $f(n)\gg\log n$ for the single-set problem, or the base $2$ in $g(k)$. The search was targeted (problem page bibliography, GST-I citation chase forward and back, $S$-unit counting literature, OEIS), not exhaustive.

2. The reduction chain, derived rather than cited

Define $g_2(k)$ = the largest $|A|$ such that there exist $b_1 \ne b_2$ with $a+b_1$ and $a+b_2$ both $S$-smooth for every $a \in A$, $|S| = k$.

(a) $g(k) \le g_2(k) + 2$. Take $A$ extremal for $g$, pick $b_1,b_2\in A$, apply the definition to $A\setminus\{b_1,b_2\}$. $\square$

(b) $g_2(k) \le N(k)$, where $N(k)$ is the maximum, over non-zero $\lambda,\mu,\nu$, of the number of coprime triples $(x,y,z)$ composed of $k$ given primes with $\lambda x+\mu y=\nu z$. This is exactly GST 1986 §3, quoted verbatim: "Each element $a_i$ yields a solution $x=a_i+b_1$, $y=a_i+b_2$, $z=1$ of the equation $x-y=(b_1-b_2)z$." The triples are coprime because $z=1$, and distinct $a_i$ give distinct triples. $\square$

(c) Therefore, conditionally:

If $\log N(k) = o(k)$, then $g(k)=\exp(o(k))$ and Erdős #126 holds.

The implication direction is one-way. #126 does not imply the $S$-unit improvement: the single-set problem constrains all $\binom n2$ sums while (b) throws away everything but $2(n-2)$ of them, so $g$ may be far below $g_2$. Indeed it must be: $g_2(k) \ge \exp(c\sqrt{k/\log k})$ by T5, while T1 already gives $g(k)\le 3\cdot2^{k-1}$. The two-set relaxation is genuinely lossy, and that loss is why the $S$-unit route ($49^k$) is worse than the 1934 elementary argument ($2^k$) despite being the deeper machinery.

(d) The room, stated precisely. For the two-set quantity the literature brackets

$$\exp\big((4+o(1))\sqrt{k/\log k}\big) \;\le\; g_2(k) \;\le\; 3\cdot7^{2k+3},$$

lower bound T5, upper bound T3+T4. Every value in that range is $\exp(o(k))$ except the top end. So: the $S$-unit route has enough headroom to settle #126 and is not obstructed by any known construction. What it cannot do is prove the 1934 authors' own conjecture $g(k)=O(k^{1+\varepsilon})$: T5 kills polynomial bounds for $g_2$ outright, so that stronger target needs an argument using more than two of the elements.

3. Does each bound help? One line each

Landing note on the T8 row (2026-10-21). Its parenthetical relies on r_186989's "every optimal witness lives below 50", which did not survive on main: admissible sets are closed under $S$-smooth dilation, so a wider box only adds larger copies of the same sets (Hunts #103 and #105), and a primitive extremal 5-set for $S=\{2,3,5\}$ reaches 475 (Hunt #107, $\{5,11,25,245,475\}$). This note withdraws that evidence only; the row's verdict is the run's.

4. Audit of hunts/r_186989/RESULTS.md

Audited as instructed, not trusted. Four findings.

1. Arm 2's headline is 1934 knowledge, verbatim. The parent reports as a measured discovery that dropping 2 from $S$ is catastrophic. Erdős–Turán's first page: "Plainly we can suppose $p_1=2$, because if the $p_1,\dots,p_k$ are all odd, we find $n\le2$." Same parity argument, same conclusion. The parent's proof is correct; its novelty is nil. Their loose thread "prove $|A|\le p-1$ when $p\notin S$" is subsumed by the Erdős–Turán lemma, which does the general-$p$ case as a halving, not a cap. (Annotated on landing, 2026-10-21: the proposed cap is in fact false for every odd $p$, because many elements may share one nonzero residue class; it was refuted on main before this arm landed, in Hunts #93, #95, #101 and #103; the witness $A=\{1,3,7,13\}$, $S=\{2,5,7\}$ used by #95 and #103 sits in r_186989's own table.)
2. Arm 3's kill is right, and stronger than the parent claims. Supermultiplicativity would give $g(k)\ge 2^k$, which must live inside T1's $3\cdot2^{k-1}-1$, a corridor of relative width $3/2$. So a composition law would not merely refute #126, it would force the 1934 bound to be sharp up to a constant, against the explicit expectation of its own authors ($n(k)=O(k^{1+\varepsilon})$, T1″). The parent should record that its arm-3 gadget contradicts a published conjecture as well as #126.
3. The measured table is below the trivial literature construction asymptotically. $A=\{1,\dots,m\}$ gives $g(k)\ge(\frac12+o(1))k\log k$ (T11), so $g$ is superlinear for free. The parent's $2,4,5,6,8,10,11$ beat $\{1,\dots,m\}$ at each measured $k$ (e.g. $k=7$: $11$ vs $9$) but the sequence carries no asymptotic information the literature lacks. Their own "seven points decide nothing" verdict stands and is if anything generous.
4. The OEIS thread is closed, negatively. 2,4,5,6,8,10,11 matches 20 OEIS sequences, all coincidental (top hit A340932, "numbers whose least prime index is odd"). A keyword search for the concept returns nothing. No tabulation of $g(k)$ exists in OEIS.

The parent's "top-ranked door is the $S$-unit counting bound, and this hunt has no purchase on it at scout cost" is correct, and §2 above makes it precise: the door is $\log N(k)=o(k)$, and the room to walk through it is proved to exist by T5.

5. What I could not settle

• Is $\log N(k)=o(k)$ conjectured true? T5 shows the count can be as large as $\exp(c\sqrt{k/\log k})$, and it is folklore-plausible that this is close to the truth, but I did not find an explicit conjecture in a source I read. Settling this costs one look at Evertse–Győry, Unit Equations in Diophantine Number Theory (CUP 2015), Ch. 6, a book I could not access in-session. This matters: if the subexponential bound is a stated conjecture of the field, then #126 is a corollary of a believed conjecture, which changes how the problem should be briefed.
• Is $\max A$ effectively bounded in terms of $S$? This is the parent's thread "the optima are all tiny". The obvious route is circular: effective $S$-unit bounds (Baker/Győry–Yu) bound $\max(x,y)$ for $\lambda x+\mu y=\nu z$ in terms of the height of the coefficients, and here $\nu = b_1-b_2$ is itself bounded only by $\max A$. Whether a non-circular effective bound exists, which would turn the parent's bounded search into an exact computation of $g(k)$, I could not decide in the time. (Annotated on landing, 2026-10-21: answered in part on main. Hunt #107 measured that the height of a primitive admissible set of sub-extremal size is unbounded at $S=\{2,3\}$, tracking the search cutoff over eleven orders of magnitude, so no such bound holds below extremal size. Whether sets of extremal size have bounded height is still open there, and Hunt #103 states uniformly bounded height as a conjecture.)
• T2 and T5 are secondary citations. Both come from Füredi–Gyarmati 2026, a peer of ours rather than the source. T5's constant $1/8$ and T2's $2^k+1$ should be checked against Compositio Math. 66 and the Erdős–Surányi book before either is leaned on.

The doors

This run mapped a ceiling (the 92-year wall at $\log n$), so it owes the list.

1. Active constraints.

2. Frozen constants.

3. Information class. Door 1 (better $N(k)$) and door 2 ($\lvert B\rvert\ge3$) both stay inside Diophantine-approximation data and are where the field's tools already point. The halving door requires reading something the current arguments do not: joint information across the $k$ primes about a single set of integers: an inverse-theorem or entropy statement, not a unit-equation count. That door is the only one that could reach the 1934 authors' own conjecture $g(k)=O(k^{1+\varepsilon})$, since T5 blocks the unit-equation route from ever proving a polynomial bound.

Sources

• T. F. Bloom, Erdős Problem #126, https://www.erdosproblems.com/126, accessed 2026-08-24.
• P. Erdős, P. Turán, On a problem in the elementary theory of numbers, Amer. Math. Monthly 41 (1934), 608–611. https://combinatorica.hu/~p_erdos/1934-03.pdf (read in full).
• K. Győry, C. L. Stewart, R. Tijdeman, On prime factors of sums of integers I, Compositio Math. 59 (1986), 81–88. http://www.numdam.org/item/CM_1986__59_1_81_0/ (read in full).
• P. Erdős, C. L. Stewart, R. Tijdeman, Some Diophantine equations with many solutions, Compositio Math. 66 (1988), 37–56.
• P. Erdős, J. Surányi, Topics in the Theory of Numbers, Springer, 2003.
• E. Füredi, K. Gyarmati, Erdős–Turán Theorem and Eulerian Integers, arXiv:2602.07545 (Feb 2026).
• J.-H. Evertse, On equations in S-units and the Thue–Mahler equation, Invent. Math. 75 (1984). GST's reference [6], the source of their Lemma 4. GST's reference list was truncated in the scan I read at entry [4], so the exact title/pages of [5] and [6] are inferred from the standard bibliography and are not citation-checked.
• F. Beukers, H. P. Schlickewei, The equation x+y=1 in finitely generated groups, Acta Arith. (1996).
• J.-H. Evertse, H. P. Schlickewei, W. M. Schmidt, Linear equations in variables which lie in a multiplicative group, Ann. of Math. 155 (2002).
• K. Győry, Some recent applications of S-unit equations, Astérisque 209 (1992), 17–38.