{"query": "Primes and zeta — Euler's bridge and the prime number theore", "count": 20, "results": [{"id": "card_alm_connection_riemann_hypothesis_open_zeta_facts", "title": "Almanac: The Riemann Hypothesis -- an open question about where the zeta function vanishes, with the established facts sealed and the conjecture marked honestly", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  The Riemann zeta function, zeta(s) = the sum of 1/k^s, is the deepest known bridge between the smooth world of analysis and the prime numbers. Several of its truths are firmly ESTABLISHED ", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "readable": false, "generated": false}, {"id": "card_theory_fundamental_theorem_of_arithmetic", "title": "Fundamental theorem of arithmetic (unique factorization)", "shelf": "theories", "surface": "secular", "snippet": "Fundamental theorem of arithmetic (unique factorization) — an engine domain that can touch it: number_theory. Calibration: seals. Every integer greater than 1 is a product of primes in exactly one way", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_src_openstax_contemporary_mathematics_key_concepts_7ea7697d", "title": "Key Concepts — Contemporary Mathematics", "shelf": "reference", "surface": "secular", "snippet": "Key Concepts\n\n3.1\n\nPrime and Composite Numbers\n\nThe natural numbers can be categorized as 1, prime numbers, and composite numbers.\n\nPrime numbers have as their only factors 1 and themselves.\n\nComposit", "authority_tier": "reference", "source": "OpenStax: Contemporary Mathematics (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_theory_peano_arithmetic_axioms", "title": "Peano arithmetic axioms", "shelf": "theories", "surface": "secular", "snippet": "Peano arithmetic axioms — an engine domain that can touch it: number_theory. Calibration: seals. Five axioms generating the natural numbers: zero is a number, every number has a successor, zero succee", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_n_639fcf317634", "title": "Primes and zeta — Euler's bridge and the prime number theorem", "shelf": "science", "surface": "secular", "snippet": "Euler tied the primes to the continuum: zeta(s) = product over primes of 1/(1-p^-s), so a\nstatement about ALL integers becomes a statement about the primes. Sealed: there are exactly 25\nprimes below 1", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_c_d721087fe2f2", "title": "The Riemann hypothesis — sealed all around,  ↔ Primes and zeta — Euler's bridge and the pri", "shelf": "connections", "surface": null, "snippet": "zeta(2)=pi²/6 (Basel), zeta(4)=pi⁴/90, zeta(6)=pi⁶/945, zeta(-1)=-1/12, the trivial zero zeta(-2)=0, and Euler's bridge zeta(2)·6 = pi².  — a concord the card itself states; mined + verified.", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "readable": false, "generated": false}, {"id": "card_n_ebe4c25c22ae", "title": "The Riemann hypothesis — sealed all around, refused at the center", "shelf": "science", "surface": "secular", "snippet": "The deepest open question about the primes, and the cleanest demonstration of the engine's\nhonesty. The FACTS seal: zeta(2)=pi²/6 (Basel), zeta(4)=pi⁴/90, zeta(6)=pi⁶/945, zeta(-1)=-1/12,\nthe trivial ", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_src_openstax_prealgebra_2_5_prime_factorization_and_the_least_co_050e33e4", "title": "2.5 Prime Factorization and the Least Common Multiple — Prealgebra", "shelf": "mathematics", "surface": "secular", "snippet": "2.5\n\nPrime Factorization and the Least Common Multiple\n\nLearning ObjectivesBy the end of this section, you will be able to:\n\nFind the prime factorization of a composite number\n\nFind the least common m", "authority_tier": "reference", "source": "OpenStax: Prealgebra (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_prealgebra_2e_2_5_prime_factorization_and_the_least_co_f844b6a5", "title": "2.5 Prime Factorization and the Least Common Multiple — Prealgebra 2e", "shelf": "mathematics", "surface": "secular", "snippet": "2.5\n\nPrime Factorization and the Least Common Multiple\n\nLearning Objectives\nBy the end of this section, you will be able to:\n\nFind the prime factorization of a composite number\n\nFind the least common ", "authority_tier": "reference", "source": "OpenStax: Prealgebra 2e (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_contemporary_mathematics_3_1_prime_and_composite_numbers_00cf5abc", "title": "3.1 Prime and Composite Numbers — Contemporary Mathematics", "shelf": "reference", "surface": "secular", "snippet": "3.1\n\nPrime and Composite Numbers\n\nFigure\n3.2\n\nComputers are protected using encryption based on prime numbers. (credit: “Data Security” by Blogtrepreneur/Flickr, CC BY 2.0)\n\nAfter completing this sect", "authority_tier": "reference", "source": "OpenStax: Contemporary Mathematics (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_theory_integers_counting", "title": "Counting & the integers (the first abstraction)", "shelf": "theories", "surface": "secular", "snippet": "Counting & the integers (the first abstraction) — an engine domain that can touch it: number_theory. Calibration: seals — arithmetic on the integers verifies exactly. Counting is the recognition that ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_alm_connection_zeta_even_powers_are_pi", "title": "Almanac: Inverse even powers fold into pi; inverse odd powers escape into mystery -- the two faces of the zeta function", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  Sum the inverse powers of the whole numbers and a startling split appears. For every EVEN power, the sum is a rational number times a power of pi: zeta(2) = 1 + 1/4 + 1/9 + ... = pi^2/6 (t", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "readable": false, "generated": false}, {"id": "card_alm_connection_euclid_euler_perfect_numbers", "title": "Almanac: Euclid-Euler: every even perfect number is a Mersenne prime times a power of two", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  A perfect number equals the sum of its own proper divisors -- 6 = 1+2+3, 28 = 1+2+4+7+14 -- and Euclid (Book IX) plus Euler proved exactly where they come from: every even perfect number i", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "readable": false, "generated": false}, {"id": "card_src_openstax_contemporary_mathematics_introduction_17741f5b", "title": "Introduction — Contemporary Mathematics", "shelf": "reference", "surface": "secular", "snippet": "Figure\n3.1\n\nEncryption of computers and messages use very large prime numbers. (credit: modification of work \"Jefferson cylinder cipher (replica)\" by Daderot/Wikimedia Commons, Public Domain)\n\nChapter", "authority_tier": "reference", "source": "OpenStax: Contemporary Mathematics (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_c_741715d501d0", "title": "The primes and the nucleus — one statistical ↔ Primes and zeta — Euler's bridge and the pri", "shelf": "connections", "surface": null, "snippet": "A prime number and a uranium nucleus carry the same deep statistical signature.  — a concord the card itself states; mined + verified.", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "readable": false, "generated": false}, {"id": "card_domchk_number_theory_claimed_prime", "title": "Number Theory: prime", "shelf": "number_theory", "surface": "secular", "snippet": "A worked check in number theory: prime.\n\nGIVEN\n  n_prime = 17\n\nCLAIMED\n  claimed_prime = True\n\nTHE ENGINE'S VERDICT\n  number_theory.primality: CONFIRMED — 17 is prime (matches claim)\n\nAND THE FALSEHOO", "authority_tier": "reference", "source": "The verifier's own documented relation, and a run this engine performed against it (deterministic; re-runnable with tools/domain_goldens.py)", "readable": false, "generated": false}, {"id": "card_n_ea9e3f380756", "title": "The primes and the nucleus — one statistical fingerprint", "shelf": "science", "surface": "secular", "snippet": "The night's deepest rhyme, and it is real, published, and unexplained. Montgomery (1973)\nand Dyson noticed that the SPACINGS between the Riemann zeta zeros follow the same distribution\nas the eigenval", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_theory_numerical_analysis", "title": "Numerical analysis (why the computer's answer is not the answer)", "shelf": "theories", "surface": "secular", "snippet": "Numerical analysis (why the computer's answer is not the answer) — an engine domain that can touch it: computer_science. Calibration: map-only — error bounds and condition numbers are computable in pr", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_src_etym_prim", "title": "prim", "shelf": "etymology", "surface": "secular", "snippet": "prim: etymology (Webster 1913) — n.: [See Privet.]; a.: [OF. prim, prin, prime, first, principal. sharp, thin, piercing, fr. L. primus first. See Prime, a.]. From Webster's Revised Unabridged Dictiona", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_domchk_atomic_claimed_valid_quantum_numbers", "title": "Atomic: valid quantum numbers", "shelf": "atomic", "surface": "secular", "snippet": "A worked check in atomic: valid quantum numbers.\n\nGIVEN\n  l = 2\n  n = 3\n\nCLAIMED\n  claimed_valid_quantum_numbers = True\n\nTHE ENGINE'S VERDICT\n  atomic.quantum_numbers: CONFIRMED — validity True matche", "authority_tier": "reference", "source": "The verifier's own documented relation, and a run this engine performed against it (deterministic; re-runnable with tools/domain_goldens.py)", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_alm_connection_riemann_hypothesis_open_zeta_facts"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}