Section14.4Exercises¶ permalink

1

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

2

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 [C.6.17] and Exercise 3.6.12.)

3

Finish proving Fact 14.3.1.

4

Find numbers writeable in two different ways as a sum of three squares.

5

Show that an odd number which is congruent to seven modulo eight may not be written as a sum of three squares.

6

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

7

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.

8

Write a program in Sage (or another language) to verify Fermat's Last Theorem for some small $x,y,z$ and $n$ .

9

Write a program in Sage (or another language) to compute $g(n)$ in the Hilbert-Waring Theorem for small $n$ .

10

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