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