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n = p + (2^a)*(3^b)

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  • Mark Underwood
    Time to procrastinate and get into some prime diversion. An observation: Finding the first odd number n 1 that cannot be written as n = p + 2^a (p prime and
    Message 1 of 1 , Oct 2, 2008
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      Time to procrastinate and get into some prime diversion. An observation:

      Finding the first odd number n > 1 that cannot be written as

      n = p + 2^a

      (p prime and a >= 0) is easy:

      n = 127.

      There's lots of them:

      127,149,251,331,337,373,509,599 ....

      But, given p prime and a,b>=0, I have yet to find an n greater than
      one and free of factors of 2 or 3 that cannot be written as

      n = p + (2^a)*(3^b)

      in more than one way.

      For instance,

      5 = 2 + 2^0*3^1
      5 = 3 + 2^1*3^0.


      35 = 3 + 2^5*3^0
      35 = 11 + 2^3*3^1
      35 = 17 + 2^1*3^2
      35 = 19 + 2^4*3^0
      35 = 23 + 2^2*3^1
      35 = 29 + 2^1*3^1
      35 = 31 + 2^2*3^0.

      I would think that the law of small numbers would finally be overcome
      at high enough n, but I'm not so sure about it.


      Mark
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