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I have been trying to figure out(prove) Goldbach Conjecture(Strong) which states: Every even integer greater than 2 can be expressed as the sum of two primes. My question I guess is general, is it wrong for me to prove something using "simple statements". Here are my statements:

  Every prime number greater than 2 is odd. 
  Every even integer is the sum of two odd numbers.
  Therefore, the conjecture is true.

In math, is it not allowed to make general statements OR must I prove this by other means like for example, proof by contradiction, direct proof etc.?

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closed as unclear what you're asking by user99914, Jendrik Stelzner, Leucippus, Lord Shark the Unknown, BCLC Aug 10 '18 at 8:23

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Every prime number greater than 2 is odd $\neq$ Every odd number greater than 2 is prime $\endgroup$ – c.. Mar 15 '17 at 1:22
  • $\begingroup$ @c.z. Hi c.z., I did not make that statement thought. Did my question seem to assume that I meant that? $\endgroup$ – Alaa Mar 15 '17 at 1:25
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    $\begingroup$ Are we down-voting because the illogical leap in the example? The question is whether such general statements are acceptable. The answer is yes, as long as the conclusion follows from the assertions, which happens to not be the case in the example. $\endgroup$ – Matt Watkins Mar 15 '17 at 1:25
  • $\begingroup$ @MattWatkins Thank you for an answer Matt. $\endgroup$ – Alaa Mar 15 '17 at 1:27
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    $\begingroup$ Or even: "Every father is male. Every king of England was male. Therefore, every king of England was a father." $\endgroup$ – Kenny Wong Mar 15 '17 at 1:29
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In general, proving something using "simple statements" is not only acceptable, but encouraged - the best proofs are the ones that use your general format. The thing is, your series of simple statements do not form a proof. In a proof, each statement must be a consequence of the one before; your first two sentences do not entail the third. The easiest way to see this is that, for example, $42$ can be written as $21 + 21$, the sum of two odd numbers. But $21$ is not prime, so $21 + 21$ isn't a way of writing $42$ as a sum of two primes. Now, we can also write $42$ as $19 + 23$, but the point is that the existence of a way to write it as a sum of two odd numbers doesn't tell us how to write it as a sum of two primes.

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The problem is that your "proof" is flawed. Just because every prime number is odd doesn't mean that every odd number is prime. If I conjectured that every even number greater than 2 was the sum of two odd square numbers (false by counter example; 6), your logic would "prove" it:

Every odd square number is (by definition), odd.
Every even integer is the sum of two odd numbers.
Therefore, the conjecture is true.
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