MATH 313 Study Guide - Midterm Guide: Fundamental Solution, Prime Factor
Document Summary
April 22, 2013: (10 points) find all solutions (x, y) to the equation x2 17y2 = 1 for which x, y > 0. In other words, nd a fundamental solution (x, y) and explain how all other solutions with x, y > 0 can be generated from it. Hint: this is the same as the same proof for quadratic residues mod a prime. if and only if. Explain brie y why this implies that z[ 1+ 13. 2: (10 points) suppose that d 5 mod 8. Show that there are no elements of norm 2 in z[ d]; deduce that the element 2 is irreducible in z[ d]. Show then that 2 | d2 d = (d + d)(d d); conclude that 2 is not prime and therefore that there is no unique factorization in z[ d]. 2 . p(cid:17) = 1 with p an odd prime.
