Math 58 - Number Theory

November 2, 2006 (corrected Nov. 6)

Homework 12 (due November 7):

 

About quadratic residues

 

1.  (same as text, section 11.2, #1)  Evaluate these Legendre symbols:

           

                (Note that answers to odd-numbered exercises are in the back of the book.  In this case, half of them are right.)

2.  (same as #2)  Show that when  p  is a n odd prime,     is

            +1  when p = ± 1 (mod 12),  and

            –1  when p = ± 5 (mod 12).

 

3.  (same as #6)  Show that there are infinitely many primes of the form 5k + 4.

            (Use  N = 5(n!)2 – 1;  you’ll need to know for which primes 5 is a QR.)

 

About Diophantine equations

 

4.  Consider this Diophantine equation:     ax + by + c = 0.    You are given integer constants           a, b, and c, and you are to find all pairs of integers (x, y) that satisfy the equation.

 

            Write a complete set of instructions for finding all solutions to this equation.

 

            You will probably need to divide into cases depending on the values of a, b, and c.

 

About Gaussian integers

 

A Gaussian integer is a complex number of the form  (a + bi)  where a and b are integers.  (We usually just write “a” for a+0i.)  The set of Gaussian integers is closed under addition, subtraction, and multiplication (defined as usual for complex numbers:  (a+bi) (c+di) = (ac-bd) + (ad+bc)i ).  The Gaussian integers satisfy all of our axioms for addition, subtraction, and multiplication, but not any of the ordering axioms.

 

As complex numbers, the Gaussian integers have lengths:  |a+bi| = sqrt(a2+b2).  For our purposes, the square roots would just be a nuisance, so we define the norm of (a+bi) by   N(a+bi) = a2 + b2.

 

5.  Show that if   (a+bi)(c+di) = (e+fi),  then  N(a+bi) times N(c+di) equals N(e+fi).

 

6.  Find all the Gaussian integers  a+bi  with…

            a.  N(a+bi) = 0

            b.  N(a+bi) = 1

            c.  N(a+bi) = 2

            d.  N(a+bi) = 3

            e.  N(a+bi) = 4

            f.  N(a+bi) = 5

            g.  N(a+bi) = 6

            h.  N(a+bi) = 7

 

7.  Which Gaussian integers have multiplicative inverses?  That is, for which Gaussian integers (a+bi) can we find (c+di) such that  (a+bi)(c+di) = (1+0i) = 1 ?   (If you’re not sure you have found all of them, consider what N(a+bi) would have to be.)

 (end)