PRP0109canonicalv1Sieve-CRT Compatibility
Sieve-CRT compatibility: n divisible by p_i iff i-th CRT residue is 0. Multiplicative sieve = product of local sieves.
Payload
Sieve-CRT Compatibility
Sieve-CRT compatibility: n divisible by p_i iff i-th CRT residue is 0. Multiplicative sieve = product of local sieves.
Sieve-CRT Compatibility
Summary
Sieve-CRT compatibility: n divisible by p_i iff i-th CRT residue is 0. Multiplicative sieve = product of local sieves.
Statement
\label{prop:sieve-crt}
An integer~$n$ is divisible by some $p_i$ ($i \le k$) if and only if
the CRT decomposition $(n \bmod p_1, \ldots, n \bmod p_k)$ has a zero
component. The multiplicative sieve equals the product of local sieves
via CRT.
\textbf{Lean:} \texttt{sieve\_crt\_compat\_3}.\quad
\textbf{Registry:} III.P42.
Proof / Justification
No immediate manuscript proof block was extracted in this pilot run.
Source Context
- Registry source:
book-03.jsonlline 256 - Manuscript source:
2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.texlines 126-134
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookIII.Spectral.SieveInfrastructure - Name:
sieve_crt_compat_3
Dependencies
- Canonical: III.D99, III.T10
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
Aliases & legacy IDs
III.P42sieve-crt-compatibilityprop:sieve-crtRelease lines
corpus_v3_workingcorpus_v2Relations
Formalized by (1)
Appears in (1)
Downstream uses (computed) (2)
Items in the corpus that reference this one via load-bearing relations. Computed from the full corpus-v3 graph at build time.
Sources
Version & History
Status disclaimer
A Corpus Item page reports the program's current internal record for this item. It does not imply external verification, scientific consensus, or final proof unless explicitly stated. Read it together with its dependencies, formalization status, and the program's overall stance.