Number Theory: In Context and Interactive

Look up the concepts of ‘Gaussian moat’, ‘Gaussian zoo’, and/or ‘Gaussian prime spiral’ and tell what you think!

Look up ‘Eisenstein integers’. Can you find any interesting theorems along these lines which they prove? What would Eisenstein primes look like? What about “Eisenstein triples”? (See [E.7.17] and Exercise 3.6.20.)

Finish proving Fact 14.3.1.

Find numbers writeable in two essentially different ways as a sum of three squares (not just \(1^2+0^2+0^2=0^2+1^2+0^2\) or even \(3^2 + 4^2 +1^2 = 0^2 + 5^2 + 1^2\)). (This was also Exercise 13.7.4.)

Show that two (separate) instances of Pythagorean triples can yield an answer to the previous exercise in a clever way. (Thanks to Samuel Paquette.)

Show that an odd number which is congruent to seven modulo eight may not be written as a sum of three squares, obviously without using Fact 14.2.1. (This was also Exercise 13.7.3.)

Research Lagrange’s four-square theorem and write an essay about it; which proof do you prefer?

Write a program in Sage (or another language) to explore which numbers may be written as a sum of two cubes, two fourth powers, and so forth.

Write a program in Sage (or another language) to verify Fermat’s Last Theorem for some small \(x,y,z\) and \(n\text{.}\)

Write a program in Sage (or another language) to compute \(g(m)\) and/or \(G(m)\) in the Hilbert-Waring Theorem for small \(m\text{.}\)

For which \(m\) do results in this chapter give us information about \(g(m)\) or \(G(m)\text{?}\) Be as specific as possible.

Look for a pattern, similar to the one we found for sums of squares, for which primes can be written in the form \(x^2+3y^2\text{.}\) Prove that the primes not of this form are impossible.

Yet another possible generalization of Pythagorean triples is to ask when the sum of two perfect powers of the same degree is a perfect square, or \(x^n+y^n=z^2\text{.}\) Explain why this is not so interesting when \(n\) is even, and why when \(n=3\) we already have seen at least one solution. Then do some experiments to conjecture whether there are solutions for prime \(n\gt 3\text{.}\) (See [E.4.23, p. 255].)