Number Theory: In Context and Interactive

1.

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!

4.

Prove Dirichlet's Theorem on Primes in an Arithmetic Progression for the case $a=2\text{.}$

5.

Find an arithmetic progression of primes of length five with less than ten between primes.

6.

Find an arithmetic progression of primes of length six or seven, starting at a number less than ten.

7.

Prove that there can be only one set of “triple primes” – that is, three consecutive odd primes.

8.

Find the value of $23\#\text{.}$

9.

Compute some twin primes greater than one thousand.

10.

Show that $\left(1-\frac{2}{p}\right)=\left(1-\frac{1}{(p-1)^2}\right)\left(1-\frac{1}{p}\right)^2\text{.}$

11.

What form must $n$ have for $n$ and $n+2$ to both not be divisible by three?

12.

Which residues modulo five must $n$ avoid for $n$ and $n+2$ to both not be divisible by five?

13.

Search a few resources to learn about “prime constellations” and write a report. The Prime Pages or Tomás Oliveira e Silva's very nice graphs of “admissible” constellations are a good place to start.

14.

Find a definition for palindromic primes (base $10\text{,}$ say) and report on the current known status. Are there infinitely many, or a way to generate them programmatically?

15.

Search a good book (see the general C.2 or specialized C.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.