Why every sequence ⌊ξ(5/2)n⌋ has infinitely many composite terms — proved by growing a small graph one prime at a time, squeezing its forced paths into words, and pruning until nothing is left. Chapter 7 extends the method to ⌊ξ(5/3)n⌋ by contracting in both directions.
This animation explains the word-labelled method from the paper Finite graph certificates for composite terms in rational floor sequences by Yuri Odagiri (Section 4, Section 5.3 and Appendix F): the proof that for every real ξ > 0, infinitely many terms of ⌊ξ(5/2)n⌋ are divisible by one of 2, 3, 7, 11, 13, 17, 19, 23, 29, 31, and hence composite. Chapter 7 proves the same for ⌊ξ(5/3)n⌋ with the primes 2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37. It continues the story of the one-step method video.
The graphs are not pictures of pre-computed results: each time it opens, the page runs stages 0 to 6 of the word-labelled computation (36,843 vertices at stage 6), and its counts agree with Table 6 of the paper. The numbers shown for stages 7 to 9, and the 210-letter word, come from the full computation reported there.
The narration was generated with Gemini 3.8 Flash TTS and plays in sync with the captions. It starts in a window-filling view; W switches to the page view with the chapter list, and F uses the whole screen. If the audio file cannot be loaded (for example when the HTML file is opened on its own), the browser's speech synthesis reads the captions instead.
Paper: DOI 10.5281/zenodo.22997709 · Sources and reproduction: github.com/ixixi/rational-floor-certificates · 日本語版 · The one-step method