Any officially registered UB undergraduate student may submit a solution for this problem. Solutions should be submitted to Dr. Eigel through the Math and Science secretary. The office is located on the first floor in Dana Hall. The deadline for submission is noon on Tuesday, December 7, 2004. A correct solution will be posted after the noon deadline. The name of the winner of the $5 prize for the best presented, complete, accurate, correct solution, if any, will be posted later.END OF SEMESTER SPECIAL! THE PRIZE IS $10 for the best presented, complete, accurate, correct solution of this problem.
PROBLEM 194 (H) Catalog
Gravity Phottle (known to his friends as Gravy) has a master’s degree in libramatics. He was hired last year by a large midwestern university to locate, identify and catalog all of the quadratics that infested the campus. He began with the so-called “basic” quadratics, those of the form ý+a?+b, where a and b are integers and where the solutions of the equation ý+ax+b=0 are integers. He used the symbol B to represent the set of all basic quadratics, and defined the category of ý+ax+b to be the integer [a, b] = a+b. His intent was to put all of the quadratics with the same category number into the same bin.
(1) Show that if [a,b] > 0, Gravy will need room for at least two quadratics in the corresponding bin.
(2) Find all positive integers N such that, if [a, b] = N, then the corresponding bin will have only two quadratics in it.