Disproving a Theorem

In the following posts, we will be looking more at how to prove different theorems. In some of these, we will have to determine if the theorem is true, then prove that our answer is correct. You can find more examples of proof writing in the Study Help category for mathematical reasoning. You can also view the available videos on YouTube.

$$n^{2}+17n+17$$ is prime.

Prove or disprove the following: $$n^{2}+17n+17$$ is prime for all $$n \in \mathbb{N}$$.

First Example

In order to determine if this theorem is true or false, we will begin by running through an example. The easiest example to run through would be $$n=1$$. We then find that $$1+17+17=35$$. However, $$35=7*5$$, so $$35$$ is not prime. We, therefore, have a counterexample to the theorem.

Proof

We claim that $$n^{2}+17n+17$$ is prime for all $$n \in \mathbb{N}$$ is a false statement. That is, we claim that there exists an $$n \in \mathbb{N}$$ such that $$n^{2}+17n+17$$ is not prime. Therefore, our proof is as follows.

Let $$n=1$$. Note that $$n \in \mathbb{N}$$. Furthermore $$1+17+17=35$$. Since $$35=5*7$$, and $$5 \neq 1$$ and $$7 \neq 1$$, we have that $$35$$ is not prime. Hence, there exists an $$n \in \mathbb{N}$$ such that $$n^{2}+17n+17$$ is not prime.

Conclusion

In this case, we were quickly able to find a counter example to the theorem. This was extremely helpful in that it saved us from having to trying to discern if there was some property that made this true, or if we just picked a few examples that worked. With the counter example in hand, the proof consisted of showing that the example we came up with satisfied the required properties needed to make the theorem false.

For more help with proofs, make sure to continue working on the other problems we have posted solutions for in Study Help. If you find these helpful, make sure to let other people know about them so that they can get help as well.

This site uses Akismet to reduce spam. Learn how your comment data is processed.