{"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_src_oeis_a003459", "title": "A003459 — Absolute primes (or permutable primes): every permutation of the digits is a prime.", "shelf": "oeis", "surface": "secular", "snippet": "Absolute primes (or permutable primes): every permutation of the digits is a prime.  First terms: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 111111111"}, {"id": "card_src_oeis_a002036", "title": "A002036 — Compressed primes: a(n) is the nearest integer to prime(n)/log prime(n).", "shelf": "oeis", "surface": "secular", "snippet": "Compressed primes: a(n) is the nearest integer to prime(n)/log prime(n).  First terms: 3, 3, 3, 4, 5, 5, 6, 6, 7, 9, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20."}, {"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_a002386", "title": "A002386 — Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.", "shelf": "oeis", "surface": "secular", "snippet": "Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.  First terms: 2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, "}, {"id": "card_src_oeis_a001275", "title": "A001275 — Smallest prime p such that the product of q/(q-1) over the primes from prime(n) to p is greater than 2.", "shelf": "oeis", "surface": "secular", "snippet": "Smallest prime p such that the product of q/(q-1) over the primes from prime(n) to p is greater than 2.  First terms: 3, 7, 23, 61, 127, 199, 337, 479, 677, 937, 1193, 1511, 1871, 2267, 2707, 3251, 37"}, {"id": "card_src_oeis_a001276", "title": "A001276 — Smallest k such that the product of q/(q-1) over the primes from prime(n) to prime(n+k-1) is greater than 2.", "shelf": "oeis", "surface": "secular", "snippet": "Smallest k such that the product of q/(q-1) over the primes from prime(n) to prime(n+k-1) is greater than 2.  First terms: 2, 3, 7, 15, 27, 41, 62, 85, 115, 150, 186, 229, 274, 323, 380, 443, 509, 577"}, {"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_oeis_a005235", "title": "A005235 — Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p.", "shelf": "oeis", "surface": "secular", "snippet": "Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p.  First terms: 3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 8"}, {"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_a001986", "title": "A001986 — Let p be the n-th odd prime. Then a(n) is the least prime congruent to 3 modulo 8 such that Legendre(-a(n), q) = -1 for all odd primes q <= p.", "shelf": "oeis", "surface": "secular", "snippet": "Let p be the n-th odd prime. Then a(n) is the least prime congruent to 3 modulo 8 such that Legendre(-a(n), q) = -1 for all odd primes q <= p.  First terms: 19, 43, 43, 67, 67, 163, 163, 163, 163, 163"}, {"id": "card_src_oeis_a001988", "title": "A001988 — Let p be the n-th odd prime. a(n) is the least prime congruent to 7 modulo 8 such that Legendre(-a(n), q) = -Legendre(-1, q) for all odd primes q <= p.", "shelf": "oeis", "surface": "secular", "snippet": "Let p be the n-th odd prime. a(n) is the least prime congruent to 7 modulo 8 such that Legendre(-a(n), q) = -Legendre(-1, q) for all odd primes q <= p.  First terms: 7, 7, 127, 463, 463, 487, 1423, 33"}, {"id": "card_src_oeis_a001990", "title": "A001990 — Let p be the n-th odd prime. a(n) is the least prime congruent to 5 modulo 8 such that Legendre(-a(n), q) = -Legendre(-2, q) for all odd primes q <= p.", "shelf": "oeis", "surface": "secular", "snippet": "Let p be the n-th odd prime. a(n) is the least prime congruent to 5 modulo 8 such that Legendre(-a(n), q) = -Legendre(-2, q) for all odd primes q <= p.  First terms: 5, 29, 29, 29, 29, 29, 29, 29, 236"}, {"id": "card_src_oeis_a001992", "title": "A001992 — Let p = n-th odd prime. Then a(n) is the least prime congruent to 5 modulo 8 such that Legendre(a(n), q) = -1 for all odd primes q <= p.", "shelf": "oeis", "surface": "secular", "snippet": "Let p = n-th odd prime. Then a(n) is the least prime congruent to 5 modulo 8 such that Legendre(a(n), q) = -1 for all odd primes q <= p.  First terms: 5, 53, 173, 173, 293, 2477, 9173, 9173, 61613, 74"}, {"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,"}]}