An integer polynomial with all pairwise sums distinct — narrowed hard, not closed
The target (erdosproblems.com/324). Does f(x) ∈ ℤ[x] exist with all pairwise sums f(a)+f(b) distinct? The trap: the obvious candidate (f(x)=x⁵) is only conjectured to work, and that conjecture is implied by Lander–Parkin–Selfridge — a separate, hard, famous open problem. We flagged that explicitly in the brief and asked for something that doesn't quietly reduce to it.
Finding 1 — killed an entire family, unconditionally. For every positive integer A, x⁵+Ax⁴ has an explicit collision. The construction: 2⁴+17⁴−13⁴−16⁴ = −10560 and 2⁵+17⁵−13⁵−16⁵ = 20 — opposite signs, so scaling all four indices by L = 528·A produces an exact collision. We checked the base identity and the general formula ourselves for A = 1, 2, 3, 5, 10 — exact match every time. The striking part: for A=1 this predicts a collision at indices (1056, 8976, 6864, 8448) — the exact same four numbers our own from-scratch brute-force search found earlier today, before we'd seen this formula. Two independent methods landing on the same answer is about as good as verification gets.
Finding 2 — why irrational works and rational can't. A clean pinning argument: for irrational α, n⁵+αn⁴ has distinct real pair sums, because a collision would force two independent integer conditions (both a 4th-moment and 5th-moment match) that already pin the pairs down to being identical. A rational coefficient collapses those two conditions back into one — which is exactly why Ruzsa's real construction needs genuine irrationality, and why no rational polynomial can simply imitate it to within a small enough error (a short limit argument rules that out flatly).
Finding 3 — the Faltings shortcut doesn't work. "The collision equation is high-degree, so genus/Faltings kills it" was our own loose intuition going in. It's dimensionally wrong: the collision locus is a three-dimensional hypersurface, not a curve. Faltings applies to curves. A real geometric approach needs an extra invariant to cut that down first.
Round 2 — how far does the construction actually reach? We noticed the original 528-identity only produces a collision for u|528 (u the leading x⁵ coefficient) — a strict subset of all possible coefficient pairs, not "every rational." We asked for the real reach. What came back: an exact dilation lemma (a base quadruple with D₅>0>D₄ reaches precisely the coefficient pairs (u,v) with u|A and B|v, for A,B derived from the quadruple) — plus two brand-new base identities outside the 528 family: (2,14,11,13) reaches u|45, 52|v; (3,28,19,27) reaches u|5, 41|v. We independently verified both identities, both explicit collisions, and a rational c=52/3 example built from the first — exact match every time. Better still: two genuine infinite families of new coefficient ratios, one accumulating toward 0, one diverging to infinity — we checked both against the general construction for five values each, all held exactly.
The sharp remaining gap, now precise. Call the set of achievable ratios 𝓡. The open question is whether 𝓡 contains every positive rational. We now know 𝓡 contains 528, 45/52, 5/41, and two infinite families — but not yet, say, the plain integer 5. One clean logical payoff: if 𝓡 = every positive rational were proven, that wouldn't just kill more individual polynomials — it would prove no rational c ever gives an eventually-Sidon sequence, full stop (a dilation-chaining argument pushes any single guaranteed collision arbitrarily far out). So closing this one gap closes the whole "rational Ruzsa" question either way.
Round 3 — from examples to a theorem. We pushed on the smallest uncovered leading coefficient, u=7 (our own N=20,000 direct search for 7x⁵+x⁴ had found nothing). What came back: a clean new identity — (17,33,23,32) gives ratio 7/67 exactly, no dilation needed — proving 7x⁵+67x⁴ collides already below N=33 (17⁵-ish scale, not the tens-of- thousands the earlier family needed). We confirmed the identity and the explicit collision (both sides equal 368,939,364) exactly.
More importantly, that specific win generalized into a real theorem: every positive integer u is reachable for at least one coprime v — not found by luck, but constructed directly via a parametrized family Q_t = (2t+1, 17t, 13t−1, 16t), choosing t=4u to force u into the numerator. We verified the family's collision holds for five values of t (1, 2, 5, 10, and 28 — the one that mechanically reaches u=7), including matching the exact D₄, D₅ values by hand for t=28.
What this doesn't do, stated precisely. This proves "every u has SOME v" — not "u=7 works with v=1 specifically," which is a genuinely harder, different question (needs the ratio to reduce all the way to B=1). The gap is now sharp: it needs simultaneous control of numerator AND denominator, not either alone. A natural next attempt — a simple affine family aimed straight at forcing B=1 — was shown NOT to work, via a clean moment/Vandermonde argument (any such family forces the whole 5th-power difference to vanish identically, killing the construction before it starts). That's a real dead end correctly identified, not just another unsuccessful search.
Sourcing honesty: the actual Ruzsa (2001) and Dubickas–Novikas (2021) papers were only reachable as abstract/preview, not full text — everything above is independently re-derived, not a reconstruction of their internal proofs. Worth saying plainly rather than implying we read papers we didn't.
Round 4: (7,1) specifically is out of reach for this whole family — a real mod-7 wall, not just a failed search. We asked directly whether the round-3 construction could hit u=7 with v=1 exactly (not just "some v"). It can't, and there's a clean reason why: write the quadruple as (2t+p, 17t, 13t, 16t) for any inward offset p — this covers the original family, every dilation of it, and a natural two-parameter enlargement. Modulo 7, the collision quantity reduces to (2t+p)⁴+t⁴, and the fourth-power residues mod 7 are only {0,1,2,4} — no two nonzero ones ever sum to 0. So the only way a multiple of 7 can show up is if both t and p are themselves multiples of 7 — which, run back through the same argument on the smaller quadruple t/7, p/7, forces them to be divisible by 7 again, and again, forever. Impossible for any actual t≥1. We verified this exactly: the mod-7 identity itself (5,000/5,000 random cases), the residue-sum fact directly, and the "7|D₄ forces 7|t and 7|p" claim against a real brute-force sweep — 6,420 genuine cases, zero exceptions. A second uniform dilation doesn't rescue it either — we checked that algebraically too: it can only inflate the denominator further, never free it back down.
What we're NOT putting on this page. The brief also argued, via a separate inequality, that the whole construction always sits at "0 < ratio < 1," which it used to claim NO version of this family can ever reach any positive integer ratio at all. We checked that claim directly against the construction's own numbers — and it doesn't hold: recomputing the actual ratio for the very quadruple that produces u=7 gives a value around 14, not something between 0 and 1. That's a real internal inconsistency, not just our own arithmetic error (we checked it two ways). So we're keeping the mod-7 result, which stands on its own and doesn't depend on that step, and dropping the broader claim rather than repeating something we caught contradicting itself.
Where this leaves 𝓡. (7,1) is provably unreachable by this entire construction family — the original quadruple, every dilation, and the natural offset enlargement. That's real, useful information: it means closing the "does 𝓡 = every positive rational" question needs a genuinely different construction, not a bigger search inside this one. It does not prove 7 is unreachable by any construction whatsoever — the brief says so itself, and we agree that's the honest scope.