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Trying to do a non-rigorous direct proof

  1. Aug 10, 2013 #1


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    x is an integer
    x is a prime number if x doesnt consist of any prime factors ≤√x

    Since (√x + 1) * (√x + 1) > √x * √x
    x must be a prime


    Whould you consider this a non-rigorous direct proof?
    If not, what does it lack?
    Is this a good approach trying to prove it?

    The proof was meant to be like this:

    Since √(x + 1) * √(x + 1) > √x * √x
    x must be a prime
    Last edited: Aug 10, 2013
  2. jcsd
  3. Aug 10, 2013 #2
    Recall that every integer >1 has a (unique) prime factorization. Now, suppose that ##x## is not prime, and further has no prime factor ##p_1 \leq \sqrt{x}## (so that ##p_1 > \sqrt{x}##); then ##x## has factorization ##x = p_1???##. What possible values can ##?## take? (Hint: what happens if ##? > \sqrt{x}##)
  4. Aug 10, 2013 #3


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    Please dont post any more clues just jet im trying to figure it out it :):)
    I dont know if this translates very well, but: "I´m going to sleep on it " ;)
    Last edited: Aug 10, 2013
  5. Aug 10, 2013 #4


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    Homework Helper

    I know you said no more hints, but this hint is just too useful not to suggest.

    Hint: write the smallest divisor (not counting 1 of course) as ##\sqrt{x} + ε##.

    Hmm, I wonder if this counts as a direct proof, probably not.
  6. Aug 11, 2013 #5


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    It is always a good idea to start with a proper statement of what it is you are trying to prove.

    "For all positive integer x, if x has no prime factors less than or equal to its square root then x is prime"

    One problem with this formulation is that it is false. The positive integer 1 is a counter-example.
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