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Math Help - Counterexample Help

  1. #1
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    Counterexample Help

    Find a counterexample to the following assertion:

    Every odd number can be expressed as the sum of a power of 2 and a prime.
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  2. #2
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    What about 5?
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  3. #3
    MHF Contributor chisigma's Avatar
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    5=2+3... and 3 is prime...

    Kind regards

    \chi \sigma
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  4. #4
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by MATNTRNG View Post
    Find a counterexample to the following assertion:

    Every odd number can be expressed as the sum of a power of 2 and a prime.
    Haha, 1.
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  5. #5
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    Quote Originally Posted by chisigma View Post
    5=2+3... and 3 is prime...

    Kind regards

    \chi \sigma
    Haha I forgot about 2^1.

    But yeah, 1 would work
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  6. #6
    MHF Contributor chisigma's Avatar
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    The smallest odd number > 1 that can't be written as the sum of a power of 2 and a prime number is 127...

    127-1= 126= 2 x 7 x 9

    127-2 = 125 = 5 x 5 x 5

    127-4 = 123 = 3 x 41

    127 - 8 = 119 = 7 x 17

    127 - 16 = 111 = 3 x 37

    127 - 32 = 95 = 5 x 19

    127 - 64 = 63 = 3 x 3 x 7

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    \chi \sigma
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  7. #7
    MHF Contributor chisigma's Avatar
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    There are seventeen odd numbers >1 and <1000 that can't be written as the sum of a power of 2 and a prime number:

    127,149,251,331,337,373,509,599,701,757,809,877,90 5,907,959,977,997...

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    \chi \sigma
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  8. #8
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    Quote Originally Posted by MATNTRNG View Post
    Find a counterexample to the following assertion:

    Every odd number can be expressed as the sum of a power of 2 and a prime.
    If you're only allowing positive numbers and addition, then 1 and 3 would fill that bill.
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