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The maximum modulus root of this polynomial is always a prime

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  • Roger Bagula
    The Perron number type of elliptical equations suggested this polynimial whose maximum modulus root is always a prime: q^2=p^3 + 2*p^2 - (3 - b[n])*p - b[n]
    Message 1 of 1 , May 1, 2005
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      The Perron number type of elliptical equations suggested this polynimial
      whose maximum modulus root is always a prime:
      q^2=p^3 + 2*p^2 - (3 - b[n])*p - b[n]
      b[n]=3*Prime[n]-Prime[n]^2
      It's kind of a cheat as the input sequence depends on the primes as well.
      The Second biggest root is mostly Abs[Prime[n]-3] except for the second
      term.
      w^2=(x-1)*(x-Abs[Prime[n]-3])*(x+Prime[n])
      The Roots->{1,0,-3}
      seem important.

      %I A000001
      %S A000001 2, 1, 1, 3, 0, 1, 5, 1, 2, 7, 1, 4, 11, 1, 8, 13, 1, 10, 17, 1, 14, 19, 1,
      16, 23, 1, 20, 29, 1, 26, 31, 1, 28, 37, 1, 34, 41, 1, 38, 43, 1, 40, 47, 1,
      44, 53, 1, 50, 59, 1, 56, 61, 1, 58, 67, 1, 64, 71, 1, 68, 73, 1, 70, 79, 1,
      76, 83, 1, 80, 89, 1, 86, 97, 1, 94, 101, 1, 98, 103, 1, 100, 107, 1, 104,
      109, 1, 106, 113, 1, 110, 127, 1, 124, 131, 1, 128, 137, 1, 134, 139, 1, 136,
      149, 1, 146, 151, 1, 148, 157, 1, 154, 163, 1, 160, 167, 1, 164, 173, 1, 170,
      179, 1, 176, 181, 1, 178, 191, 1, 188, 193, 1, 190, 197, 1, 194, 199, 1, 196,
      211, 1, 208, 223, 1, 220, 227, 1, 224, 229, 1, 226, 233, 1, 230, 239, 1, 236,
      241, 1, 238, 251, 1, 248, 257, 1, 254, 263, 1, 260, 269, 1, 266, 271, 1, 268,
      277, 1, 274, 281, 1, 278, 283, 1, 280, 293, 1, 290, 307, 1, 304, 311, 1, 308,
      313, 1, 310, 317, 1, 314, 331, 1, 328
      %N A000001 Absolute values of the three roots of p^3 + 2*p^2 - (3 - b[n])*p - b[n]:
      b[n]=3*Prime[n]-Prime[n]^2
      %C A000001 The maximum modulus root of this polynomial is always a prime.
      %F A000001 b[n]=3*Prime[n]-Prime[n]^2
      a(n) = Abs[root[i,m]]/.p^3 + 2*p^2 - (3 - b[n])*p - b[n]
      %t A000001 b= Table[3* Prime[n] - Prime[n]^2, {n, 1, Digits}]
      a = Abs[Flatten[Table[p /. Solve[p^3 + 2*p^2 - (3 - b[[n]])*p - b[[n]] == 0, p][[i]], {n, 1, Digits}, {i, 1, 3}]]]
      %O A000001 1
      %K A000001 ,nonn,
      %A A000001 Roger L. Bagula (rlbagulatftn@...), May 01 2005
      RH
      RA 69.233.150.223
      RU
      RI



      --
      Roger L. Bagula email: rlbagula@... or rlbagulatftn@...
      11759 Waterhill Road,
      Lakeside, Ca. 92040 telephone: 619-561-0814}
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