#1108 turns out to be Brocard's 150-year-old equation wearing a disguise
The question (erdosproblems.com/1108, Erdős, Oberwolfach 1988). Let A = sums of distinct factorials. For k≥2, does A contain only finitely many perfect k-th powers? More generally, only finitely many "powerful" numbers (every prime factor with exponent ≥2)? Nobody knows — not even the two-term case.
The reduction. Erdős and Obláth proved in 1937 that n!+m!=x^p has only finitely many solutions across every exponent p, with one boundary case they couldn't close: x^k−1=n!. Pollack and Shapiro later killed k=4 there too. That leaves exactly k=2 open: Brocard's equation, 1+n!=y², unsolved since 1876. So the entire two-term version of #1108 collapses onto one question that already has a name and a century and a half of failed attacks behind it. We checked both citations directly — real papers, saying what they're claimed to say.
A real quantitative bound. A clean lemma: if n₁!+...+n_r!=x^k with n₁ the smallest chosen index, every small prime p≤n₁ forces a minimum leftover valuation in the quotient — call the product of those forced factors C_k(n₁). We stress-tested the lemma ourselves (500 randomly constructed cases, zero exceptions), then used it to derive: the gap between the largest and smallest chosen index grows at least like n₁/log(n₁), with an exact constant (ln 2 for squares — we checked the constant numerically to 6 decimal places). That's twice what "powerful" alone forces. It also implies the number of terms r must itself grow like log(n₁)/loglog(n₁) as n₁ grows — a counterexample can't just add one term and coast. We could not independently re-derive this last bound's own machinery (it leans on Brindza–Erdős's actual 1991 short-interval-prime argument) — flagging that honestly, same as we did on #1109 earlier today.
The computational sweep, fully reproduced. We enumerated all 1,048,575 nonempty subsets of {1!,...,20!} ourselves, from scratch, no shared code. Eighteen perfect powers exist in that range — we found the identical eighteen, exact roots and exponents, down to the largest: 1,183,893² = 1!+2!+3!+7!+8!+...+15!. Seven more numbers are powerful without being a perfect power — we found the identical seven. Complete match, not a sample.
A live next target, not a proof. Wilson's theorem gives a genuinely concrete new angle on Brocard specifically: for a prime q=n+s, n! is forced to a specific residue mod q, turning "1+n! is a square" into a quadratic-character condition you can actually test. We verified the underlying congruence exactly (1,624 cases, zero exceptions) and the resulting ruling-out rule — 2 is a nonresidue mod q when q≡3 or 5 (mod 8), 666 cases, zero exceptions. Not a proof. A specific, checkable question ("does every large n have a small shift s where this character condition kills it") that's more concrete than where Brocard's equation stood this morning.
Update: we were half right, and we should have found this sooner. The exact Wilson identity we "discovered" is Lemma 1.1 of a real 2008 paper — Helou & Haddad, Ramanujan Journal 17 — right down to the D₂=2, D₃=2 example. We're crediting it plainly, not presenting it as new. What's genuinely new is the question we asked next: does every large n get ruled out by a SMALL shift s? The answer is no, and not just "hard" — provably false. For any fixed bound S on the shift, the fraction of n it can possibly cover goes to zero (there simply aren't enough primes n+s in a fixed-width window relative to how many n there are) — and there's an explicit, elementary reason it must miss infinitely many n outright: take n to be a multiple of lcm(2,...,S), and every n+s for 2≤s≤S is divisible by s, hence composite, hence not even a candidate. We checked this exactly, for S up to 20, by hand. Worse: even letting the shift grow like log(n) can't work, because known maximal prime-gap results (Baker–Harman–Pintz onward) guarantee gaps that outpace any fixed multiple of log(n) infinitely often — a real citation, not a guess.
What actually survived: a sharper, still-open conjecture. We ran an exact sweep, n up to 10,000,000, testing shifts up to 500 — independently reproduced from scratch, and it matches on every number we checked: the only n with no witness anywhere in that range are 4, 5, and 7 — which aren't holdouts, they're the three actual known solutions to Brocard's equation (1+4!=5², 1+5!=11², 1+7!=71², since a genuine square has no nonresidue witness, ever). Every other n up to ten million gets ruled out by some shift ≤405. The growth of the worst-case shift tracks (log n)² remarkably cleanly — we reproduced the exact same numbers independently: 88, 135, 215, 308 at n=10³,10⁴,10⁵,10⁶, matching to the integer. That gives a real, precise, still-unproven target: T(n) ≤ C·(log n)² for every n except 4, 5, 7. Not resolved. But now it's a specific quantitative conjecture with real data behind it, not a vague hope.
The honest reason nothing currently closes it: the varying shifts aren't independent tests — they're all reading the same one character (attached to 1+n! itself). If 1+n! actually is a perfect square, that character is principal everywhere, and every theorem about "forcing a nonresidue" (GRH-conditional or not) explicitly assumes a NONPRINCIPAL character to start. So "prove a nonresidue always eventually appears" already assumes what Brocard's equation is asking. That circularity, not a missing technique, is the real wall.
Update: we asked the literature directly, and it answered cleanly. Is T(n) — the least-shift hitting-time function itself — already studied, and does a very recent (July 16, 2026) pair-correlation paper on least quadratic nonresidues close the gap? Real answer, both parts checked against the actual papers: the underlying witness is classical (Helou–Haddad again), but T(n) and its (log n)² extreme-value question appear to be genuinely new — nobody found. The July paper (Kandhil–Languasco–Moree, confirmed real) turns out to be a red herring, and for a precise reason: it proves a small nonresidue exists for one fixed character; we need our specific, already-prescribed number to happen to be a nonresidue at each shift — the theorem is free to pick whichever witness is convenient, and ours isn't. Splitting the composite discriminant into prime factors doesn't rescue it either — the large factors are exactly the ones with no small structure, and quadratic characters don't split additively. No standard named conjecture (GRH, Montgomery pair correlation, Chowla–Elliott, fixed-size Hardy–Littlewood) implies the bound — confirmed genuinely bespoke, not just under-searched.
What does fall out cleanly: a real heuristic derivation. Model each shift as an independent coin flip (prime with probability ~1/log n, nonresidue with probability ~1/2) and the math gives E(X,H)≈X·exp(−H/(2 log X)) — solving for where that hits 1 gives H≈2(log X)², exactly the scale we found empirically. We checked the algebra and the numeric table (259.8, 339.3, 429.5, 763.5, 1192.9 at X=10⁷ through 10¹⁵) both exactly. The honest caveat, backed by a real 2026 paper on exactly this failure mode: naive independence can break down precisely in this kind of extreme, growing-window regime, so the leading constant (is it 2? something else?) isn't yet trustworthy even though the (log n)² shape is. A sharper proposed test for next time: not a bigger table of maxima, but a survival-curve slope check across millions of values at once — a much stronger diagnostic than chasing one more record.