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From bad to worse...from good to Angelic

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  • Roger L. Bagula
    I actually didn t find bad primes defined in any number theory way... they were associated with a Calabi-Yau based L-function in a paper with no real good
    Message 1 of 1 , May 23, 2008
      I actually didn't find "bad primes" defined in any number theory way...
      they were associated with a Calabi-Yau based L-function in
      a paper with no real good definition.
      So I did a web search and an OEIS search and some experimentation.
      A little thought of the geometric average basis of the defintion gave
      me an higher order "bad" I call "worse".
      An a good I call "Angel".


      %I A000001
      %S A000001 19, 43, 47, 61, 73, 79, 83, 89, 109, 113, 137, 139, 157,
      167, 181, 197, 199,
      211, 229, 233, 239, 241, 271, 281, 283, 293, 313, 317, 353, 359, 383, 389,
      401, 439, 443, 449, 463, 467, 503, 509, 521, 523
      %N A000001 Worse primes<-bad primes (A130903):
      If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2] < 0,
      Prime[n]]
      %F A000001 a(n) = If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2] < 0,
      Prime[n]]
      %t A000001 Flatten[Table[If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2]
      < 0,Prime[n], {}], {n, 3, 100}]]
      %Y A000001 Cf. A130903
      %O A000001 1
      %K A000001 ,nonn,
      %A A000001 Roger Bagula and Gary W. Adamsom (rlbagulatftn@...),
      May 23 2008
      RH
      RA 192.20.225.32
      RU
      RI


      %I A000001
      %S A000001 5, 7, 11, 13, 17, 23, 29, 31, 37, 41, 53, 59, 67, 71, 97,
      101, 103, 107,
      127, 131, 149, 151, 163, 173, 179, 191, 193, 223, 227, 251, 257, 263, 269,
      277, 307, 311, 331, 337, 347, 349, 367, 373, 379, 397, 409, 419, 421, 431,
      433, 457, 461, 479, 487, 491, 499, 541
      %N A000001 Angel primes<-good primes (A028388):
      If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2] > 0,
      Prime[n]]
      %C A000001 This sequence set is a next level higher of good and bad
      primes.
      The bad primes come from modular forms and L-Series theory.
      http://en.wikipedia.org/wiki/Arithmetic_of_elliptic_curves:
      Quote:
      "Reduction mod p
      Reduction of an abelian variety A modulo a prime ideal of (the
      integers of) K - say, a prime number p - to get an abelian variety Ap
      over a finite field, is possible for almost all p. The 'bad' primes,
      for which the reduction degenerates by acquiring singular points, are
      known to reveal very interesting information. As often happens in
      number theory, the 'bad' primes play a rather active role in the theory.
      Here a refined theory of (in effect) a right adjoint to reduction mod
      p - the Néron model - cannot always be avoided. In the case of an
      elliptic curve there is an algorithm of John Tate describing it.
      For abelian varieties such as Ap, there is a definition of local
      zeta-function available. To get an L-function for A itself, one takes
      a suitable Euler product of such local functions; to understand the
      finite number of factors for the 'bad' primes one has to refer to the
      Tate module of A, which is (dual to) the étale cohomology group
      H1(A), and the Galois group action on it. In this way one gets a
      respectable definition of Hasse-Weil L-function for A. In general its
      properties, such as functional equation, are still conjectural - the
      Taniyama-Shimura conjecture was just a special case, so that's hardly
      surprising.
      It is in terms of this L-function that the conjecture of Birch and
      Swinnerton-Dyer is posed. It is just one particularly interesting
      aspect of the general theory about values of L-functions L(s) at
      integer values of s, and there is much empirical evidence supporting it."
      %D A000001 http://en.wikipedia.org/wiki/Arithmetic_of_elliptic_curves
      %F A000001 a(n) = If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2] <>0,
      Prime[n]]
      %t A000001 Flatten[Table[If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2]
      > 0,Prime[n], {}], {n, 3, 100}]]
      %Y A000001 Cf. A130903,A028388
      %O A000001 1
      %K A000001 ,nonn,
      %A A000001 Roger Bagula and Gary W. Adamsom (rlbagulatftn@...),
      May 23 2008
      RH
      RA 192.20.225.32
      RU
      RI


      --
      Respectfully, Roger L. Bagula
      11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814
      :http://www.geocities.com/rlbagulatftn/Index.html
      alternative email: rlbagula@...
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