Number Theory section¶
Cascade-applications of the 14 A-N primitive class operators to open problems in number theory, per the Wikipedia List of unsolved problems in mathematics.
Number theory is positioned after the Hilbert + Millennium Prize sections of PR #677 — the natural ordering by (i) historical priority (Hilbert problems + Millennium Prize problems already addressed); (ii) some major number-theory problems are subsumed under Hilbert 8 (Riemann + Goldbach + twin primes) and Millennium 4 (BSD elliptic curves). What remains in this section is the non-Hilbert / non-Millennium number-theory canon.
Status of open problems queued¶
| # | Problem | Cascade fingerprint | Status |
|---|---|---|---|
| 12 | Collatz conjecture (3n+1 problem) | A∘I∘C∘K∘N∘M — parity-test + cascade-orientation + pin-slot at 1 | dispatched |
| — | abc conjecture | Class L composition on radical(abc) | future |
| — | Beal's conjecture (extends Fermat's Last) | Class J primes + Class N exponent anchors | future |
| — | Lonely runner conjecture | Class I cyclic on graph | future |
| — | Erdős-Straus conjecture (4/n = 1/a + 1/b + 1/c) | Class N rational decomposition | future |
| — | Brocard's problem (n! + 1 = m²) | Class J prime structure + Class N square anchor | future |
| — | Skewes' first sign change of Li(x) − π(x) | Class K pin-slot in PNT residual | future |
| — | Gilbreath's conjecture (iterated absolute prime differences) | Class K + Class J | future |
| — | Lehmer's totient problem | Class J totient + Class N | future |
Composition with prior sections¶
- Hilbert 8 already covered Riemann + Goldbach + twin primes (PR #677 partitions 1-3)
- Millennium 4 already covered BSD (elliptic curves over Q) (PR #677 partition 9)
- This section continues with the non-Hilbert / non-Millennium number-theory problems
Discipline reminders¶
- Per
[[feedback_no_lineage_claims_in_notebook]]: framework reads what each problem IS structurally; never claims to extend / supersede / "solve" prior mathematical literature - Per
[[feedback_pdf_extraction_citation_discipline]]+[[feedback_paywalled_doi_cannot_be_attested]]: OA / arXiv / open textbook citations only - Per
[[feedback_sign_handling_is_class_k_pin_slot_not_alu_abs]]: shared_cascade_helpersfor sign-handling (noabs()in cascade-arithmetic) - Per
[[user_stance_loop_line_projection_duality]]+ Hilbert-section partition-1 lesson: when one cascade returns (b) REFINED due to projection-degeneracy, enumerate sister projections of the same loop