Today I spent two hours reading a section of Analysis with an Introduction to Proof and doing the associated exercises. I’m feeling better than I was for most of last week, though the mental weather is still not entirely clear.
Part of the section I read explained the inadequacy of examples, even quite a few examples, in proving general statements. As an illustration, it offered the statement that $n^2+n+17$ is prime for every natural number $n$. This statement is true for values of $n$ up to 15, but is not true for 16 or 17. The latter, in particular, could be predicted without trying each value of $n$, since $17^2$, $17$, and $17$ are clearly all divisible by a common factor (meaning that their sum must also be divisible by that factor).
Later, an exercise asked me to find a counterexample to the statement that $3^n+2$ is prime for every natural number $n$. This had me wondering whether there was a way to predict a counterexample, as there had been in the earlier case.1 I could not think of one, however, so I answered the question by trying each $n$. It turns out that I didn’t have to go far, as $n=5$ breaks the pattern. I still wonder if there is a better way, though.
Thank you to all of you who have commented on my posts recently. It really helps me stay committed to my project. I will try to reply soon.
- To do. ↩︎
Glad to see you back
Thanks. I’m glad to be back, though I wish I were as well mentally as I was near the beginning of this project.
In the case of your polynomial, you saved work by looking at it mod 17. Could you at least save computational labor by doing something similar here?
I had some thoughts along those lines that I didn’t pursue. I will try to see if they lead anywhere.
Examples have utility. Even quite a few examples do little to prove general statements, but it only takes a single counterexample to disprove one.
I could not get anywhere on the question you posed.
True, examples are powerful to disprove.