Homework problems assigned

Set 1:   sect 1.3 #1,3,6,8
         sect 1.2 #8,19,20,21

Set 2:   sect 1.4 #6,7,40
         sect 1.5 #5,6,16,36,44

Set 3:   sect 3.2 #14, 15, 18, 35
         sect 3.3 #2,6,10
**       sect 14.2 #16, 20  *** <-- removed from assignment

Set 4:   sect 3.4 #2, 4, 14, 19
         sect 3.5 #2, 6, 11, 12, 18, 22, 30, 36

Set 5:   sect 4.1 #13, 22, 33
         sect 4.2 #2, 8, 12, 14

Set 6:   sect 4.3 # 4, 12, 14
         sect 4.4 # 4, 8
         sect 4.5 # 2, 3
*This HW is due Friday Mar 13 (the last day before Spring Break)*

Ooookay, Nature threw us a curve ball, and now everything is messed up.
The Grader and I will figure out what to do with both paper and
electronic copies of HW 6. Meanwhile, here are some questions you 
can start to work on for HW7. Let's set the due date as Friday, Apr 10.
You will have to submit these electronically through Canvas.

Set 7:   sect 6.1 # 4, 10  (Please use Wilson's and Fermat's theorems, not brute force)

      Then here are a few problems that pick up on this theme of primality-testing:
      A  Use Fermat's Theorem to prove that N=2^11-1 is composite.
	   (Possible hint: what's the fastest way you could compute 3^32 mod N ?)
           NOTE: if the computation starts to be annoying, try N=253=2^8 - 3  instead;
                 in that case, consider computing  47^32 mod N .
      B  Find all solutions to the congruence x^2=1 mod 2^n, for n=3, 4, 5.
      C  Find all solutions to the congruence x^2=1 mod 3^n, for n=1, 2, 3.
      D  Show that if N is not a power of a prime, then there exists an integer k
	   for which  k^2  is congruent to  1  mod  N  , but  k  is not congruent to  +1
	   nor to  -1  mod  N  . (Example: 41^2 = 1 mod 28, yet I know 41 isn't
	   congruent to +1 mod 28 because it's congruent to -1 mod 7; and I know
	   41 isn't congruent to -1 mod 28 because it's congruent to +1 mod 4 ...)
      E  Show that if  k  is an integer for which  k^2=1 mod N,  and  k  is
           not congruent to +1 nor to -1 mod  N,  then  gcd(k-1, N)  is a
	   nontrivial factor of  N .

      I can give you more exercises if you like that explore this theme: this leads
      to reasonably efficient ways to test to see if a number is prime, and if not,
      to discover its factors. But that's completely optional.


Set 8: 9.1 # 2, 12
       9.2 # 6, 16
       9.3 #10, 12