{"id":"palomar/PALOMAR-2026-10-27-000017","url":"https://github.com/jasondavies/pierce-length-bound/tree/6da6511cea9f8526054dd5ad60d77164f45de52e","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-27-000017-v1.json","repo":"https://github.com/jasondavies/pierce-length-bound","tags":["math.NT","msc:11A67","msc:11N37","msc:11H06"],"score":0,"title":"Bounds for the iteration x ↦ a mod x","trust":"high","domain":"math.NT","source":"palomar","authors":["Jason Davies"],"reviews":0,"subject":"math.NT","summary":"For integers 0 0, including the external coefficient-uniform affine-curve theorem and the effective Hilbert specialisation sieve, and its rational instance 4317/14677. It also proves the exterior-lattice exponents 471/1559 - 6 sqrt(105)/7795 and 153/520, the joint-invariant exponents (1310-sqrt(655))/4365, 2201/7480 and 128/435 and the earlier exponents 45/152, 8/27, 11/37, 23/77, and 3/10. It also proves a bound O(a^(1/4) sqrt R), where R minus one counts rises in the quotient increments, and positivity and integrality of the orbit's Hankel invariants. A separate conditional theorem gives…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-27","reactions":{},"formalized":true}