MATH 250   Number Theory and Mathematical Reasoning

Questions about the positive integers 1, 2, 3 . . . have fascinated people for thousands of years. The ancient Greeks noted the existence of right triangles with sides of integral length, corresponding to equations such as 3² + 4² =5² and 5² + 12² = 13². Is there a way of describing all such "Pythagorean Triples?" As another example, the integer 6 is called “perfect” because 6=1+2+3, so that 6 is the sum of its positive factors. There are currently 51 known perfect numbers. Do the perfect numbers follow any pattern? Are there infinitely many of them? Much of this course focuses on such questions as a means for introducing students to the spirit and methods of modern mathematics. Other course content deals with the question of extending the list 0,1,2,3,... so that it will include the number of elements in any finite set, but also the number of elements in any infinite set. The emphasis throughout is on developing the ability to construct logically sound mathematical arguments and communicate these arguments in writing.