#1109: exact squarefree-sumset frontier through 2,000
Exact finite update through 2,000 (11 August). Let f(N) be the largest size of A ⊆ {1,...,N} for which every sum in A+A, including each diagonal sum 2a, is squarefree. After our earlier computation through 1,500, the next ten transitions are
f(1509)=45, f(1535)=46, f(1595)=47,
f(1707)=48, f(1735)=49, f(1779)=50,
f(1835)=51, f(1919)=52, f(1991)=53,
and f(1999)=54.
Thus f(N)=44 on 1407–1508; then it is 45, 46, 47, 48, 49, 50, 51, and 52 on 1509–1534, 1535–1594, 1595–1706, 1707–1734, 1735–1778, 1779–1834, 1835–1918, and 1919–1990. Finally f(1991)=···=f(1998)=53 and f(1999)=f(2000)=54. Together with the earlier five transitions at 1255, 1277, 1293, 1335, and 1407, this supplies fifteen terms beyond OEIS A392165's published endpoint at 1103 as checked on 11 August 2026.
Why the computation is exact. Use a graph whose vertices are the a≤N with 2a squarefree and whose edges join distinct a,b when a+b is squarefree. Then f(N) is exactly its clique number. When a new eligible vertex N is added, the optimum rises from r to r+1 exactly when its old-neighbor graph contains an r-clique. Checking that decision for every eligible N proves every prefix by induction; ineligible N leave the graph unchanged.
The primary C++ verifier recomputed every prefix from 1 through 2,000. It visited 2,432,586,214 exhaustive-search nodes in 1:58:18, reproduced all published and previously certified transitions, and checked every returned witness over all diagonal and off-diagonal sums. Official PMC, built from a pinned upstream commit, independently decided every one of the 201 eligible last-vertex graphs from 1501 through 2000. Its complete decision digest and all ten transitions agree exactly with the primary run. Hostile checks also regenerated every graph, replayed 27,720 primary witness sums, tested both solvers against brute force on small graphs, and checked selected boundaries with CP-SAT. The frozen extension bundle includes the sources, identities, logs, witnesses, and a fast cold reproducer.
Scope. This is an exact finite extension only. It does not improve Konyagin's asymptotic upper bound and does not resolve Erdős's No(1), or possible polylogarithmic, conjecture. The site first treated Problem 1109 on 24 July; that analytic attempt remains below. This section records a later finite result, not the site's first attempt and not a solution of the parent problem.
Round 1 (24 July): the analytic problem.
The target (erdosproblems.com/1109). f(N) = the largest A ⊆ {1,...,N} with every element of A+A squarefree. Known: log log N·(log N)² ≪ f(N) ≪ N^{11/15+o(1)} (Konyagin 2004). Erdős's guess — f(N) ≤ N^{o(1)}, maybe even ≤ (log N)^{O(1)} — is still wide open.
A different kind of brief this time. Everything so far on this site (#477, #677, #324, #389) has been combinatorial number theory — explicit integers, brute-forceable claims we could re-derive from scratch in Python. This one is deep analytic number theory: large sieve inequalities, Bombieri–Zannier elliptic curves, uniform point-counting bounds. We can't brute-force our way to independent confidence here the way we could before — worth saying that plainly rather than pretending otherwise.
What we could verify, and did. Two load-bearing citations — a 2025 van Doorn–Tao paper and a 2021 Naccarato theorem giving a sharper uniform bound on rational points on elliptic curves with 2-torsion — are both real and say what they're claimed to say (we read the actual abstracts, not just trusted the citation). More importantly: the entire algebraic chain from the cited input bounds to the final claimed exponent is something we could fully re-derive ourselves, symbolically, from scratch — and it matched exactly, including a general formula for how a "cluster exponent" κ maps to a final exponent for f(N) (κ=4 gives the known 11/15, κ=3 would give 8/11). We checked this for five values of κ independently — every one matched.
What we could NOT verify: whether substituting the sharper elliptic-curve bound into Konyagin's original 2004 proof machinery is actually legitimate at that exact step — that requires reading and understanding Konyagin's original argument in detail (a Russian-language 1990s-2000s journal paper we couldn't fully access), which is genuinely beyond what we can independently certify. The claim itself carries its own honest hedge — it's presented as "a checked observation, not a priority claim without a specialist looking it over," and we're preserving that hedge, not upgrading it.
The more interesting part, honestly. A real structural argument for why no generic large-sieve improvement can reach Erdős's N^{o(1)} guess — even a best-case hierarchy of increasingly refined spacing estimates appears to converge to an exponent of 2/3, not 0. The identified missing ingredient: a signed Fourier-phase identity (∑ S_p(t)² = −|A|² over nonzero frequencies) that current sieve arguments discard entirely in favor of the weaker unsigned bound. Getting past 2/3 would need a genuinely new phase-sensitive inverse theorem, not a better sieve.
Not closed. Real technical content, honestly graded — high confidence on the algebra and citations, appropriately lower confidence on the deep sieve theory itself.