# Erdos

- 2026-09-23 [Verified] The two-part obstruction at twelve digits: for each of the `30` base sets in `[3, 10]` minimal for `sigma > 1` and `gcd = 1`, the least `sum(M_1(D) below N)/N` over terms `N` in `[10^12, 10^30]` lies in `[1.105953, 1.537469]`, truncated, so any two disjoint parts of the terms have one with `Delta >= 0.23 N` at that `N`. Witness: `lab/rs/mixed-powers`, verb `route 10 12`
- 2026-09-23 [Verified] Erdos problem 124 at the first three levels up to base 10: every base set `D` in `[3, 10]` with `sigma(D) > 1` and `gcd(D) = 1` has `sum_(d in D) P(d, k)` cofinite for `k = 1, 2, 3`, by the surplus certificate on the `30` minimal base sets, `21` of them also at `k = 4` and `5` at `k = 5`, with a `2^32`-bit array per cell in `2.9` s. Witness: `lab/rs/mixed-powers`, verb `census 10 5 32 4`
- 2026-09-23 [Verified] Erdos problem 124 at the first two levels up to base 12: every base set in `[3, 12]` with `sigma > 1` and `gcd = 1` has `sum_(d in D) P(d, k)` cofinite for `k = 1, 2`, over the `103` minimal base sets, whose `sigma` lies between `2311/2310` and `8/7`; the last cell, `{3, 6, 9, 10, 12}` at `k = 2`, needs a `2^34`-bit array and has largest non-sum `1473914231`. Witness: `lab/rs/mixed-powers`, verbs `census 12 5 32 4` and `cell 3,6,9,10,12 2 34`
- 2026-09-23 [Verified] The largest integer that is no sum of distinct powers `3^j, 4^j, 5^j` with `j >= k` is `79, 77613, 4330731, 1075364603` at `k = 1, 2, 3, 4`, with `11, 1128, 45704, 1785062` positive non-sums up to it; for `3^j, 4^j, 6^j` it is `986, 242113, 58941162` at `k = 1, 2, 3`; a plain knapsack over every term up to `4 F` returns the same largest non-sum and count on nine cells. Witness: `lab/rs/mixed-powers`, verbs `census 10 5 32 4`, `cell 3,4,5 4 34` and `control`
- 2026-09-23 [Verified] Fan 2026 Theorem 1.5, read at source, proves the set of powers of a finite `D` strongly complete when `D` splits into two parts of `sigma >= 1` and one of `gcd = 1`, no two elements powers of one integer, and Bergelson and Simmons 2017 Theorem 1.23, read at source, the same with three parts of `sigma >= 1`, stating they can neither prove nor disprove the conjecture of Burr, Erdos, Graham and Li; the Erdos problem 124 page, last edited before Fan's paper, cites neither, and both theorems need `sigma(D) >= 2`, so no theorem whose proof is read on the page settles a base set with `sigma(D) < 2`; the page's credit of `{3, 4, 7}` to Burr, Erdos, Graham and Li rests on a paper not read. Witness: `erdos.md` Problem 124: what is in print
- 2026-09-23 [Conjecture] The powers `3^j, 4^j, 5^j` with `j >= k` are complete at every level `k`, the least base set outside every theorem whose proof is read on the page; checked at `k <= 4`. Witness: `lab/rs/mixed-powers`, verbs `census 10 5 32 4` and `cell 3,4,5 4 34`
- 2026-09-23 [Proved] The surplus certificate for Erdos problem 124, the surplus bound being the per-part computation of Fan 2026 Section 7: for a finite base set `D` of integers `>= 3` with `sigma(D) = sum 1/(d - 1) > 1` and a level `k >= 1`, if the sums `P_n` of the `n` least terms of the multiset `M_k(D)` of powers `d^j`, `j >= k`, contain every `x` with `T <= x <= S_n/2`, `T - 1` is missing, `T <= a_(n+1)`, and `a_(m+1) <= S_m - 2T + 1` for every `m >= n` with `(sigma(D) - 1) a_(m+1) < C_k(D) + 2T - 1`, `C_k(D) = sum d^k/(d - 1)`, then every integer `>= T` is a sum of distinct terms and `T - 1` is the largest that is not. Witness: `erdos.md` The surplus certificate, "Proved (the certificate)"
- 2026-09-23 [Proved] Exact in the gap: when `1 < sigma(D) < 2`, `M_k(D)` is complete if and only if the surplus certificate exists at some `n`, so the certificate search halts exactly on the complete levels and returns the largest non-sum; the proof takes Dirichlet scales `N` where every base has its least power `>= N` in `[N, beta N)`, `sigma(D) beta < 2`. Witness: `erdos.md` The surplus certificate, "Proved (exact in the gap)"
- 2026-09-23 [Proved] Completeness of `M_k(D)` passes to every base set containing `D` and every level below `k`. Witness: `erdos.md` The surplus certificate, "Proved (monotone)"
- 2026-09-23 [Proved] The two-part route is dead below two: for finite `D` with `sigma(D) < 2` and any `k >= 0`, no two disjoint infinite sub-multisets of `M_k(D)` both have `Delta(B) = sup_(b in B) (b - sum_(b' in B, b' < b) b')` finite, so the criteria of Fan 2026 Theorem 2.2 (two such parts) and Bergelson and Simmons 2017 Theorem 2.1 (three) apply to no `M_k(D)` with `sigma(D) < 2` under any partition of the terms, while the whole of `M_k(D)` has `Delta <= C_k(D)` once `sigma(D) >= 1`. Witness: `erdos.md` The two-part route is dead below two
- 2026-09-23 [Proved] Bergelson and Simmons 2017 Main Theorem 2.1, read literally without infinite parts `B_1, B_2, B_3`, fails: at `D = {5, 6}`, `k = 1`, the parts `{5}, {6}, {25}` have `Delta` finite, the remaining powers meet the divergence and residue conditions, and the sums of distinct powers number at most `x^(log_5 2 + log_6 2) < x^0.82` up to `x`; its proof and Fan 2026 Theorem 2.2 use infinite parts. Witness: `erdos.md` Problem 124: what is in print
- 2026-09-23 [Verified] Sets and multisets differ on Erdos problem 124 where two bases are powers of one integer: the powers of `{3, 5, 6, 9}` from `k = 1` have largest non-sum `22` as the multiset of the site's question, while as a set of powers `649` is no sum and every integer from `650` to `2 10^4` is. Witness: `lab/rs/mixed-powers`, verbs `cell 3,5,6,9 1 20` and `set 3,5,6,9 1 20000`
- 2026-09-23 [Verified] Fan 2026 Theorem 1.5 needs `sigma(D) > 2`, its third part being nonempty with `gcd = 1`, and Bergelson and Simmons 2017 Theorem 1.23 needs `sigma(D) > 3`; the surplus computation behind the certificate first appears in Bergelson and Simmons Section 2.7, read at source. Witness: `erdos.md` Problem 124: what is in print
- 2026-09-23 [Verified] The seed: the `33` terms of `floor(2 (5/3)^n)` up to `2^26` open `31` windows `a_(n+2) > S_n` and every one holds a non-sum; the largest non-sum up to `2^25` is `23559582`. Witness: `lab/rs/mixed-powers`, verb `graham 2 5 3 26`
- 2026-09-23 [Verified] The power multisets run no window chain in `[10^6, 10^30]`: `M_k({3, 4, 5})` opens a window at `27` of `123` terms at each `k = 1..4`, `M_1({3, 4, 6})` at `41` of `120`, `M_1({3, 5, 6, 7})` at `5` of `142`, and in none of the six does an upper window `(S_m, a_(m+2))` meet the lower window `(S_n - a_(n+1), a_(n+2) - a_(n+1))` of a later window. Witness: `lab/rs/mixed-powers`, verb `windows`
- 2026-09-23 [Verified] The certificate for `{3, 4, 5}` at `k = 5` does not fit a `2^34`-bit array; the base interval sits at the term `a_n = 243, 177147, 9765625, 3486784401` at `k = 1..4`. Witness: `lab/rs/mixed-powers`, verbs `census 10 5 32 4`, `cell 3,4,5 4 34` and `cell 3,4,5 5 34`
- 2026-09-23 [Refuted] L, the statement that every multiset with linear surplus, `S_n - a_(n+1) >= eps a_(n+1) - C` for some `eps > 0` and a constant `C`, and `H_1(M) = {theta in R/Z : sum_(a in M) norm(a theta) < infinity} = {0}` is complete, which would have settled Erdos problem 124 above `sigma = 1`. Witness: `A = {2 F_m - 1 : m >= 2} = 1, 3, 5, 9, 15, 25, 41, ...`, `a_(n+1) = a_n + a_(n-1) + 1`, has `S_n - a_(n+1) = a_n - n - 2`, `H_1(A) = {0}` since `norm(theta) <= norm(a_(n+1) theta) + norm(a_n theta) + norm(a_(n-1) theta)`, and windows `a_(n+2) - S_n = n + 3` that nest from step to step two later, carrying the non-sum `2` into every window; the `38` terms to `2^27` open `36` windows, all holding a non-sum, and the non-sums below `10^6` are the chains `2, 7, 22, 63, ...` and `2, 11, 36, 103, ...`. Witness: `erdos.md` The missing lemma, and `lab/rs/mixed-powers`, verb `refute 27`
- 2026-09-23 [Proved] The window: for a sorted multiset with `a_(n+2) > S_n`, the sums in `(S_n, a_(n+2))` are exactly `a_(n+1)` plus the sums of the first `n` terms in `(S_n - a_(n+1), a_(n+2) - a_(n+1))`, so an integer of the lower window missing from the sums of the first `n` terms is carried to a non-sum in the upper one. Witness: `erdos.md` The missing lemma, "Proved (the window)"
- 2026-09-23 [Proved] Surplus does not heal: the set `floor(2 (5/3)^n)`, `n >= 1`, has `S_n - a_(n+1) >= a_(n+1)/2 - n - 5`, linear surplus since `a_(n+1)` grows geometrically, and is incomplete, its windows opening at every `n` and each upper window `(S_(n-2), a_n)` lying inside the lower window of step `n` from `n = 4` on, so one non-sum in a window recurs in every other window; this reproves at one point the incompleteness of Graham for `t > 1`, `alpha >= max(2/t, phi)`, quoted in Fan 2026. Witness: `erdos.md` The missing lemma, "Proved (surplus does not heal)"
- 2026-09-23 [Verified] `g_3(n) = 1, 3, 8, 22, 60, 168` at `n = 1..6` for Erdos problem 817, recomputed by an exhaustive top-down search over sets with distinct ternary sums, pruned by the previous values and by `sum a_i^2 >= (9^n - 1)/8`; the witness at `n = 6` is `{107, 145, 159, 162, 164, 168}`, a second extremal set beside the discussion thread's `{107, 145, 159, 162, 166, 168}`. Witness: `lab/rs/mixed-powers`, verb `ternary 6 200 8`
- 2026-09-23 [Verified] `g_3(7) <= 474`, past the `504` on the Erdos problem 817 thread: `{302, 409, 447, 459, 465, 466, 474}` has `2187` distinct ternary sums and its `128` subset sums hold no non-trivial 3-term progression, checked from the definition; with `g_3(n + 1) <= 3 g_3(n)` this gives `g_3(n)/3^n < 0.216736` for every `n >= 7`, against the thread's `0.2305`. Witness: `lab/rs/mixed-powers`, verb `band 419 475 70 8`, and the crate test on the set
- 2026-09-23 [Verified] No admissible 7-set for Erdos problem 817 with largest element `N` in `[419, 473]` has its six largest elements in `[N - 100, N]`, so a set beating `474` has its second-smallest element below `N - 100`. Witness: `lab/rs/mixed-powers`, verb `band 419 473 100 8`