I'll give you one the grad students asked me studying for their algebra qual, @Balarka. Let $p$ be prime, $n<p$. Show that the Sylow $p$-subgroups of $S_{np}$ are all abelian.
I tried reading some probabilistic number theory once but as the notes grew more fond of mentioning ergodic theory in later chapters, I thought better of it.
I really had a terrible test yesterday. But I get an extra one, sirens went off in the middle. Plus it wasn't that terrible, I got the answers right, just the questions were ugly..
too much effort is wasted trying to simplify expressions in High School. What is simpler is subjective, and rewriting a number 1000 times does not indicate any understanding. you might even make errors when "simplifying" and lose points.
@BalarkaSen I think so. If there are infinitely many twin primes, it's unlikely that the sum would be rational, but it might happen. Proving $S\notin\mathbb{Q}$ would prove the twin prime conjecture.
But more to anything, the main reason is probably that infinite induction doesn't work.
For example $\{1\}$ has finite cardinality. Assume $\{1, 2, \cdots, k\}$ has finite cardinality. $\{1, 2, \cdots, k + 1\}$ has finite cardinality, hence $\Bbb N$ is finite.
