{"id":"hexagon/2610.00665","pdf":"https://hexagonmath.org/pdf/2610.00665v1","url":"https://hexagonmath.org/abs/2610.00665v1","tags":["msc:11G05","msc:11G50","msc:11B25","msc:14J27"],"score":0,"title":"A Linear In Rank Bound For Integral Arithmetic Progressions On Elliptic Curves","domain":"math.AG","origin":"ai-generated","source":"hexagon","authors":["Jacob Crim"],"reviews":0,"subject":"math.AG","summary":"Every known bound on the length N of an x-progression on an elliptic curve of rank r has the shape N ≤ C^{1+r}. We prove a bound that is linear in the rank for integral progressions, with an error depending only on the height of the model. Let y^2 = x^3 +a_2x^2 +a_4x+a_6 be an integral model and H_f = 1+|a2|+|a4|+|a6|. For every ε > 0, every integral x-progression of length N satisfies N ≤ (3/2 +ε)s+C_{ε} (1+log(H_f))^{1/2}, where s ≤ r is the rank of the span of the progression. In particular N ≤ (3/2 +o(1))s whenever log H_f = o(N^2). The proof has two halves. The local theory consists of…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}