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Factoring 16**137 - 1 : second thoughts

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  • Kermit Rose
    16**137 - 1 = 2**(4*137) - 1 = (2**(2*137) - 1) (2**(2*137) +1 ) = (2**137 - 1) (2**137 + 1) (2**(2 + 2 * 136) + 1) = (2**137 - 1) (2 * 137 + 1) ( 4 * 2**
    Message 1 of 1 , Feb 1 4:01 PM
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      16**137 - 1
      = 2**(4*137) - 1
      = (2**(2*137) - 1) (2**(2*137) +1 )
      = (2**137 - 1) (2**137 + 1) (2**(2 + 2 * 136) + 1)
      = (2**137 - 1) (2 * 137 + 1) ( 4 * 2** 136 + 1)
      = (2**137 - 1) (2 * 137 + 1) (4 * 16**34 + 1)

      = (2**137 - 1) (2 * 137 + 1)
      ( 2 * 2**68 - 2 * 2**34 + 1) (2 * 2**68 + 2 * 2**34 + 1)


      I had thought that I had programmed already the factoring of

      (4 * 16**34 + 1)
      = ( 2 * 2**68 - 2 * 2**34 + 1) (2 * 2**68 + 2 * 2**34 + 1)

      But apparently there is a bug in my code for the case
      that the larger fourth power is even.

      But even this does not get the largest factor < 10**42.

      What else do I need to know?

      Kermit
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