Math 311M   Fall 2017   Homework

  Hw 1   Exercises 1.1 on p. 15:
              Practice problems (do not submit):  Ex. 1(i), 4, 5;
              Submit 8/30:  Ex. 1(iii, iv),  3,  6.   Show all your work in Ex. 1.

  Hw 2   Read:  Binomial Theorem p. 18-20.
              Exercises 1.2 on p. 23:
              Practice problems (do not submit):  Ex. 1, 2, 6, 8(i);
              Submit 9/6:  Ex. 3,  5,  8(ii).

  Hw 3   Read:  p. 32-35.  Correction: In Goldbach's conjecture it is "every even integer greater than 2".
              Exercises 1.3 on p. 35:
              Practice problems:  Ex. 1, 2, 3, 4, 5;
              Submit 9/13:  Ex. 3 (for 136 and 150 only),  6,  8,  9 (for a, b ›1).
              Hints.   Ex. 6.   Prove an equivalent statement:
              If n is not prime (i.e. either 1 or composite), then 2n-1 is not prime.
              For composite n, express 2n-1=2mk-1 = (2m)k-1 as a product of two factors, each ›1.
              To do it, recall or figure out how to factor ak-1.
              Ex. 8.   Note that every odd number is either of the form 4k+1 or of the form 4k+3.

  Hw 4   Exercises 1.4 on p. 48:
              Practice problems:  Ex. 1, 2, 3;
              Submit 9/20:  Ex. 3(ii, iv),  5,  6 (with p›2), 7.
              Ex. 3.  If the inverse does not exist explain why; if it does show how you found it.
              Ex. 5.  What can m2  be congruent to mod 8?
              Ex. 6.  Assume that p›2. In this problem x denotes a congruence class, x=[a]p.
              So we need to solve the equation  a2≡1 mod p.
              Ex. 7.  Consider p=2, 3 first.  Let p be at least 5. What does Ex.6 tell us about the inverses of [2]p, .., [p-2]p ?

  Hw 5   Practice problems:  Ex. 1.5: 1(i,ii,iv,vi), 2(i,ii), 3;    Ex. 1.6: 2(i)
              Submit 9/27:  Ex. 1.5: 1(iii,v), 5;    Ex. 1.6: 1(i), 2(iii)
              Show all your work in each of the problems. The solutions must be logically structured and easy to follow.
              In Ex. 5 in 1.5, write the corresponding congruences and find their simultaneous solution.

              More practice problems:  Ex. 1.6: Ex. 2, 5, 6(i,ii).

  Hw 6   due 10/11    pdf   

  Hw 7   due 10/18    pdf