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Thread: Example as a proof

  1. #1
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    Question Example as a proof

    Hi,

    I hope someone can help. I'm trying to determine whether the following statements can be proven merely by an example? It doesn't matter whether the statements are actually true, let's just assume for this exercise that they are true. As follows:

    Statement 1: "Powers of primes increase without bond"
    It seems that this would not be proven solely with just an example since the idea of "without bond" suggests an infinite series and not just a single value. Sounds vague what I'm saying, but hopefully I'm on the right track with my thinking. Please let me know!

    Statement 2: "There are prime numbers of the form 3^k + 1"
    I would think that you could prove this by example (assuming that the statement is true, even though I secretly know it is false) by plugging in whatever value for k results in the expression being a prime number. Is this right for me to suggest?

    Statement 3: “Not all integers q which cause a^(q−1) to have remainder 1 after dividing by q, are prime”
    I would think that you could prove by example for the same reasons that I provided for Statement 2.

    Statement 4: “There are infinitely many prime numbers”
    It appears that this could not be proven by example since the idea of infinity cannot be proven in just an example, and therefore neither can you prove infinity in the context of prime numbers. Right?

    Really really would appreciate critique and feedback on my thinking! I look forward to a response.

    - Olivia
    Last edited by otownsend; Sep 13th 2017 at 07:31 PM.
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  2. #2
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    Re: Example as a proof

    1) cannot be proven by example because you can't provide an example of an infinitely large prime

    2) can be proven by example. You only need find one prime of the form $3^k+1$ to prove they exist

    3) assuming it's true this can be proven by example. Simply find an integer that meets the requirement.
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  3. #3
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    Re: Example as a proof

    was 4) there before?

    4) what you say about 4 is correct, for the same reason as 1.
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  4. #4
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    Re: Example as a proof

    If you say that there are n or more instances of X, where n is a positive, finite number, you can prove it using n examples. If the number is left vague, such as "some" or "at least one," you can prove it using one example.
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    Re: Example as a proof

    Statements of the form "for all x" can never be proved by an example. Statements of the for "there exist" or "at least one" can be proved by an example showing one such object.
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    Re: Example as a proof

    Quote Originally Posted by HallsofIvy View Post
    Statements of the form "for all x" can never be proved by an example. Statements of the for "there exist" or "at least one" can be proved by an example showing one such object.
    The statement, "For all $x \in \{ 1,2 \}$, $x<3$," can be proven by example.
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  7. #7
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    Re: Example as a proof

    Two examples! But "For all x\in (1, 2), x< 3" cannot be.
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    Re: Example as a proof

    Gotcha ! Thanks for all the help
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