claimroutineinferred_dep
Distinct primes meet resampling hypotheses
Distinct odd primes with meet resampling
chooses the prime side lengths for the CRT step
Statement (paper)
For all sufficiently large , one can choose pairwise distinct odd primes with and for every , such that satisfies , and, with , one has , , and . Thus every hypothesis of Lemma holds, and the setup condition of Lemma holds; moreover the machine finds these primes deterministically in time .
In plain English
For large , one can deterministically choose distinct odd primes with and , whose product lies in . With they satisfy , and , so the one-dimensional and tensor resampling lemmas apply, and the search takes time .
Why it's here
This turns the abstract resampling lemmas into a concrete setup by supplying prime lengths slightly below the power-of-two-sized . The conditions follow from the gap and the parameter bounds of the previous claim, and ensures the prime-length transform still covers the needed size. The later CRT address-map step uses these primes, and the sub-linear search time keeps the construction within the overall time budget.
Classified routine:
This parameter-selection step adapts the Harvey–van der Hoeven (2019/2021) choice of distinct primes slightly below power-of-two lengths, using prime number theorem / Bertrand-type density in short intervals and deterministic trial-division search. The inequalities follow from the bounds of the previous claim by direct calculation.