RESULTS: Erdős #126, the S-unit arm (support run `d5d5ccae`)

Hunt, not a result. Nothing here is evidence for or against RH (docs/08). Support run for 0897a5a7. Arm label: sunit-equations.

Verdict, in one line: the arm delivers deliverable (3) of the three the brief allowed, with a clean version of deliverable (2) attached. The injectivity lemma the brief asked for exists, is two lines long, and reduces Erdős #126 to a bound on the number of solutions of a two-variable S-unit equation. The strongest theorem that actually applies is Evertse's, and it is weaker than the 1934 elementary bound by a factor of about 24.5^k. At k = 7 the machinery says g(7) ≤ 6.98 × 10^14, the 1960 elementary argument says 128, and the true solution count of the very equation the machinery is bounding is 96.

Reproduce with python3 hunts/support_d5d5ccae/probe.py (~100 s, stdlib only, writes results.json).

Notation follows hunts/r_186989/RESULTS.md: S is a set of k primes, A a finite set of distinct positive integers with every a + b (a ≠ b) supported on S, and g(k) the maximum |A|.

1. The lemma, which is the reusable part

Lemma (two-base-element injectivity). Let A be admissible for S, and let a > b be any two elements of A. Put d = a - b > 0. Then

the map x ↦ (a + x, b + x) injects A \ {a, b} into Sol_S(d) := {(U, W) : U - W = d, U, W positive S-smooth}.

Proof. For x ∈ A \ {a, b} both a + x and b + x are off-diagonal sums, so both are S-smooth, and their difference is d. Injectivity is immediate: x = U - a. ∎

Corollary. |A| ≤ 2 + min_{a > b ∈ A} N_S(a - b), where N_S(d) = #Sol_S(d), and hence

g(k) ≤ 2 + max_A min_{a>b∈A} N_S(a-b) ≤ 2 + max_{d ≥ 1} N_S(d).

N_S(d) is finite by Mahler's theorem, so this is a genuine finite bound.

Implication direction, checked explicitly. A bound on the number of solutions of the S-unit equation gives a bound on g(k) gives Erdős #126. The converse does not hold: g small says nothing about N_S. This arm therefore points the right way, unlike arm 3 of r_186989, which turned out to be a refutation route wearing the clothes of a proof route.

Two elementary facts fall straight out and are worth recording because they are free. gcd(a+x, b+x) divides d, so Sol_S(d) decomposes as ⨆_{g | d, g S-smooth} {g · (primitive solution of u - w = d/g)}. And d = 1 is the Størmer case: N_S(1) is exactly the number of pairs of consecutive S-smooth integers, a quantity Lehmer computed exactly by Pell descent.

2. What happened to the cross-ratio / three-base-element version

The brief suggested fixing several base elements. Fixing three, a, b, c, gives the linear identity

(b-c)(a+x) + (c-a)(b+x) + (a-b)(c+x) = 0,

a three-term equation with fixed nonzero coefficients in three S-unit unknowns. It is strictly worse, for a reason worth stating: it is a linear consequence of the two-element statement (subtract pairs and you recover (a+x)-(c+x) = a-c), so it carries no extra information, while the coefficients b-c, c-a, a-b are differences of elements of A and are not S-smooth in general. That is the load-bearing obstruction for the whole arm: the coefficient primes are outside S and their number is not bounded by k, so the S-cardinality-based theorems (Evertse) cannot absorb them, and one is pushed onto the rank-based theorems (Beukers–Schlickewei, Evertse–Schlickewei–Schmidt), which cost more.

The two-element version dodges this exactly once: U - W = d is (1/d)U + (-1/d)W = 1, and Evertse's theorem allows arbitrary fixed coefficients a, b ∈ K* while charging only for S. So the whole arm's quantitative content sits in the two-element lemma, and the fancier gadgets are a downgrade. Cross-ratios of four elements were also tried and give products of differences ((a+b)(c+d) - (a+c)(b+d) = (a-d)(c-b) and its two companions), which reintroduces the same non-S coefficient problem.

3. The quantitative accounting, done honestly

The strongest applicable theorem. Evertse (1984), as quoted in Beukers–Schlickewei §1: for a, b ∈ K* fixed, d = [K:ℚ], s = #S counting all infinite places, the equation ax + by = 1 in S-units of K has at most 3 · 7^{d+2s} solutions. Over ℚ with S = k primes plus the archimedean place, d = 1, s = k+1, so

N_S(d) ≤ 3 · 7^{2k+3} and therefore g(k) ≤ 2 + 3 · 7^{2k+3} ≈ 1029 · 49^k.

The rank-based alternative is worse here. Beukers–Schlickewei (1996), Theorem 1.1: if G is the ℚ-closure of a finitely generated subgroup of (ℂ*)² of rank r, then x + y = 1 has at most 2^{8r+8} solutions in G. Our solution pairs lie in the group generated by Γ_S × Γ_S (rank 2k) and the point (1/d, -1/d) (one more), so r ≤ 2k+1 and the bound is 2^{16k+16}. That is 65536^k, against Evertse's 49^k. Rank-based bounds are the right tool for the three-element gadget and the wrong tool here.

The comparison that decides the arm.

So: standard S-unit bounds are quantitatively too weak, by a factor ≈ 24.5^k against the best of them. This is deliverable (3) of the three the brief named, and it is the honest answer. No improved bound on g(k) comes out of this arm.

4. How much of that slack is the theorem and how much is the route

This is the question the brief's "calculate the dependence on k honestly" is really asking, and it is answerable by measurement, because N_S(d) can be counted exactly inside a box.

probe.py enumerates all S-smooth integers up to 10^14 for S = the first k primes and counts Sol_S(d) exactly inside that box. Counts inside a box are lower bounds on the true N_S(d).

Oracle check first. For d = 1 the measured counts are 1, 4, 10, 23, 40, 68, 108, 167 for k = 1..8, which reproduces Lehmer's published Størmer table exactly. The box is large enough and the counter is right.

Column 4 is 2 + min_{a>b} N_S(a-b) evaluated at the maximal witness r_186989 found for that k, with every witness re-verified from scratch by trial division. Read that column carefully: it is not an upper bound on g(k). It bounds |A| for any A containing that particular witness, and even then only as a measurement, since a box-truncated count is a lower bound on N_S and the bound needs an upper one. It is here to show how tight the lemma is at the sets that actually achieve the measured maxima.

Three things it says.

1. The lemma is not the problem. Its content at the extremal witnesses runs 3, 6, 10, 23, 35, 67, 96 against 2^k = 2, 4, 8, 16, 32, 64, 128. It is the same order as the classical bound and at k = 7 it is below it. The route is not throwing away an exponential factor; the theorem is.
2. The theorem is the problem, by about thirteen orders of magnitude at k = 7. Evertse bounds a quantity whose measured value is 96 by 7.0 × 10^14.
3. The truth about these equations looks polynomial in k, not exponential. The Størmer counts N_S(1) fit ≈ k^{3.2} over k = 6..10 (Lehmer's table continues 241, 345). max_{d≤40} N_S(d) grows faster, ratio about 1.6 per step in the measured range, which is not separable from polynomial at eight points and we are not claiming it is.

5. What would suffice, and whether it is plausible

From the corollary: any improvement of the base 49 = 7² in Evertse's ℚ-case bound to a base below 2 already beats the 1934/1960 elementary bound, and

N_S(d) = exp(o(k)) uniformly in d ⟹ g(k) = exp(o(k)) ⟺ Erdős #126.

Is that plausible? The best known lower bound for the number of solutions of a two-variable S-unit equation is Erdős–Stewart–Tijdeman (1988): exp((4+o(1))·(s/log s)^{1/2}), which is itself exp(o(s)). So nothing known rules out exp(o(k)), the measured Størmer data is polynomial, and the gap between exp(c√(s/log s)) and 3·7^{2s} is the actual open problem this arm lands on. It is a well-studied problem with its own literature, which is strictly better than landing on nothing, and it is also a problem that has resisted since 1984, which is why this is a report and not a result.

6. What this arm could not settle

• No improved bound on g(k). None. The lemma is sound and the input it needs does not exist at the required strength.
• No proof that max_d N_S(d) is subexponential in k, which is what the route needs. Eight measured points cannot separate k^{3.2} from 1.6^k.
• No control over which d an adversarial A presents. The corollary minimises over the pairs of A, which is favourable, but proving that some pair of any admissible A has small N_S(a-b) is an unsolved combinatorial step, and it is the one concrete missing lemma this arm can name (§7).
• Whether this reduction is already in the literature. Given Győry–Stewart–Tijdeman (1986) proved the c·log|A| prime-factor bound for two sets by S-unit methods, it very likely is, in some form. We did not find a citation in the time available and do not claim originality for the lemma.

7. The one concrete missing lemma

Wanted. A constant c and a function φ(k) = exp(o(k)) such that every admissible A with |A| > c contains two elements a > b with N_S(a-b) ≤ φ(k).

With it, the corollary gives g(k) ≤ 2 + φ(k) and Erdős #126 follows. The measured data is consistent with it in the strongest available way: every witness r_186989 found for k ≥ 4 contains the consecutive pair {1, 2} or another pair with d in the small-N_S range, which is why column 4 tracks N_S(1) so closely. Whether that is forced or accidental is not settled here.

The doors

This run measured a ceiling (the quantitative strength of the S-unit route), so it owes the door list.

1. Active constraints at the optimum.

2. Frozen-constant inventory.

3. Information class. Doors 2, 3 and 4 stay inside the data this arm already reads (smoothness of integers in a box) and can only sharpen the measurement, never produce the bound. Door 1 (arity with S-unit coefficients) stays inside too. The bound itself requires reading more: a genuine improvement to the quantitative theory of S-unit equations, which is Diophantine approximation and the Subspace Theorem, not enumeration. That is the same conclusion r_186989 reached from the other side, now with the number attached: the required improvement is from base 49 to base < 2.

Loose threads

• Is the two-element lemma in Győry–Stewart–Tijdeman (1986)? Their c·log|A| theorem for two sets is proved by S-unit methods and is the same wall. First step: read that paper's §2 and check whether |A| ≤ 2 + N_S(d) appears; if it does, this arm's contribution is the measurement, not the lemma.
• max_d N_S(d) for d beyond 40, and its true growth rate. First step: extend the sweep to d ≤ 10^4 at k ≤ 8 using the same smooth-number set; it is one loop and no new machinery.
• Every measured extremal witness contains a pair with tiny N_S(a-b). Why it might matter: it is exactly the missing lemma of §7 in observed form. First step: search for an admissible A at k=5,6 all of whose differences have large N_S, by adding the constraint to r_186989's clique search.