Explain why, to show that any number can be written as a sum of three primes, it suffices to prove Conjecture 22.3.8.
2.
In Subsection 22.1.3 a statement is made about residue classes \([a]\) such that \(nk+a\) can be a perfect square. What is another name for such \(a\text{?}\)
Also, the claim is made that, “In the two examples we showed graphically, only \(4k+1\) and \(8k+1\text{,}\) respectively, are possible perfect (odd) squares.” Either prove this claim or find the reference for when that is proved in the book.
3.
What ‘teams’ would you expect to be in the lead long-term for a modulo ten prime race? Why? Compute a value where the ‘wrong’ team is in the lead, if you can!
(base \(10\text{,}\) say) and report on the current known status. Are there infinitely many, or a way to generate them programmatically?
15.
Search in a good book (see the general E.2 or specialized E.4 references) or the internet for an amazing fact about primes. Describe it in a way your classmates (or peers, if you’re not in a course) will understand.