{"query": "Primes and zeta — Euler's bridge and the prime number theore", "count": 20, "results": [{"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."}, {"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"}, {"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 "}, {"id": "card_src_oeis_a000961", "title": "A000961 — Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).", "shelf": "oeis", "surface": "secular", "snippet": "Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).  First terms: 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49."}, {"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."}, {"id": "card_src_oeis_a001259", "title": "A001259 — A sequence of sorted odd primes 3 = p_1 < p_2 < ... < p_m such that p_i-2 divides the product p_1*p_2*...*p_(i-1) of the earlier primes and each prime factor of p_i-1 is ", "shelf": "oeis", "surface": "secular", "snippet": "A sequence of sorted odd primes 3 = p_1 < p_2 < ... < p_m such that p_i-2 divides the product p_1*p_2*...*p_(i-1) of the earlier primes and each prime factor of p_i-1 is a prime factor of twice the pr"}, {"id": "card_src_oeis_a005094", "title": "A005094 — Number of distinct primes of the form 4k+1 dividing n minus number of distinct primes of the form 4k+3 dividing n.", "shelf": "oeis", "surface": "secular", "snippet": "Number of distinct primes of the form 4k+1 dividing n minus number of distinct primes of the form 4k+3 dividing n.  First terms: 0, 0, -1, 0, 1, -1, -1, 0, -1, 1, -1, -1, 1, -1, 0, 0, 1, -1, -1, 1, -2"}, {"id": "card_src_oeis_a000879", "title": "A000879 — Number of primes < prime(n)^2.", "shelf": "oeis", "surface": "secular", "snippet": "Number of primes < prime(n)^2.  First terms: 2, 4, 9, 15, 30, 39, 61, 72, 99, 146, 162, 219, 263, 283, 329, 409, 487, 519, 609, 675, 705, 811, 886, 1000."}, {"id": "card_src_book_38000", "title": "Bridge; its Principles and Rules of Play with Illustrative Hands and the Club Code of Bridge Laws — J. B. (Joseph Bowne) Elwell", "shelf": "gutenberg", "surface": "secular", "snippet": "Bridge; its Principles and Rules of Play with Illustrative Hands and the Club Code of Bridge Laws, by J. B. (Joseph Bowne) Elwell. Subjects: Bridge whist. Read the full text (public domain): https://w"}, {"id": "card_src_oeis_a005850", "title": "A005850 — Primes p such that the NSW number A002315((p-1)/2) is prime.", "shelf": "oeis", "surface": "secular", "snippet": "Primes p such that the NSW number A002315((p-1)/2) is prime.  First terms: 3, 5, 7, 19, 29, 47, 59, 163, 257, 421, 937, 947, 1493, 1901, 6689, 8087, 9679, 28753, 79043, 129127, 145969, 165799, 168677,"}, {"id": "card_src_oeis_a000882", "title": "A000882 — Number of twin prime pairs <= product of first n primes.", "shelf": "oeis", "surface": "secular", "snippet": "Number of twin prime pairs <= product of first n primes.  First terms: 0, 1, 4, 15, 69, 468, 4636, 57453, 896062, 18463713, 425177757, 11997649372, 385088898632, 13280323588034, 509456736126003, ."}, {"id": "card_src_oeis_a002371", "title": "A002371 — Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).", "shelf": "oeis", "surface": "secular", "snippet": "Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).  First terms: 0, 1, 0, 6, 2, 6, 16, 18, 22, 28, 15, 3, 5, 21, 46, 13, 58, 60, 33, 35, 8, 13, 41, 44."}, {"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"}, {"id": "card_c_9aaf105f7dd8", "title": "Primes and zeta — Euler's bridge and the pri ↔ The Riemann hypothesis — sealed all around, ", "shelf": "connections", "surface": null, "snippet": "the Riemann zeros control exactly how the actual count wobbles around the smooth estimate  — a concord the card itself states; mined + verified."}, {"id": "card_src_oeis_a001031", "title": "A001031 — Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).", "shelf": "oeis", "surface": "secular", "snippet": "Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).  First terms: 1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, 2, 4, 3, 4, 4, 3, 3, 5, 4, 4, 6."}, {"id": "card_src_oeis_a002071", "title": "A002071 — Number of pairs of consecutive integers x, x+1 such that all prime factors of both x and x+1 are at most the n-th prime.", "shelf": "oeis", "surface": "secular", "snippet": "Number of pairs of consecutive integers x, x+1 such that all prime factors of both x and x+1 are at most the n-th prime.  First terms: 1, 4, 10, 23, 40, 68, 108, 167, 241, 345, 482, 653, 869, 1153, 15"}, {"id": "card_src_book_38120", "title": "Bridge Axioms and Laws — J. B. (Joseph Bowne) Elwell", "shelf": "gutenberg", "surface": "secular", "snippet": "Bridge Axioms and Laws, by J. B. (Joseph Bowne) Elwell. Subjects: Bridge whist. Read the full text (public domain): https://www.gutenberg.org/ebooks/38120"}, {"id": "card_src_book_69", "title": "The 32nd Mersenne Prime Predicted by Mersenne — David Slowinski", "shelf": "gutenberg", "surface": "secular", "snippet": "The 32nd Mersenne Prime Predicted by Mersenne, by David Slowinski. Subjects: Numbers, Prime; Number theory. Read the full text (public domain): https://www.gutenberg.org/ebooks/69"}, {"id": "card_src_book_51216", "title": "Advanced Bridge; The Higher Principles of the Game Analysed and Explained — J. B. (Joseph Bowne) Elwell", "shelf": "gutenberg", "surface": "secular", "snippet": "Advanced Bridge; The Higher Principles of the Game Analysed and Explained, by J. B. (Joseph Bowne) Elwell. Subjects: Bridge whist. Read the full text (public domain): https://www.gutenberg.org/ebooks/"}, {"id": "card_src_book_58096", "title": "A Trip to the Chain-Bridge, Near Bangor, and Other Parts of North Wales — Anonymous", "shelf": "gutenberg", "surface": "secular", "snippet": "A Trip to the Chain-Bridge, Near Bangor, and Other Parts of North Wales, by Anonymous. Subjects: Wales, North -- Description and travel; Menai Bridge (Wales). Read the full text (public domain): https"}]}