A course in elementary number theory concerns itself primarily with simple, arithmetical manipulations of counting numbers: 1, 2, 3, and so forth. These numbers hold a great deal more secrets than one might imagine at first. Despite their apparent simplicity, some of the hardest, most difficult problems in mathematics arise from the study of the theory of numbers. Grade-schoolers can comprehend questions whose solutions evade centuries of investigation. Entirely new fields of mathematical study have grown from research into these questions made possible by number theory.One of the fundamental objects of study involves the prime numbers. Nearly every integer can be built by multiplying prime numbers, which means that we can solve a large number of problems simply by thinking about them in terms of prime numbers.Even when a number n is not prime, we can restrict our arithmetic to a set of numbers relatively prime to n and recover many properties of a prime number. A signal for this is related to another important tool of number theory, the greatest common divisor (gcd).The gcd allows us to solve linear Diophantine equations, which look like the linear equations you studied in precalculus, but restrict their solutions to integer values. Related to the study of linear Diophantine equations is a clockwork mathematics called congruence, where you can assert with a straight face that, for example, 1 + 1 = 0 and not be thrown out of the room.As the course draws to its finale, we combine prime numbers, relatively prime numbers, and congruence to explain how a deceptively simple problem – factoring an integer into two primes – is in fact so difficult that it can guarantee the security of internet communication, including the credit card number you type when you make an online purchase!Along the way, we take a few detours for the sake of sightseeing. We examine some problems that fascinated the ancient Greek cult of Pythagoreans – perhaps unto death! This segues nicely into classes of numbers that are obtained easily from the integers, but have some unsettling properties. A recurring theme will focus on how these other classes of numbers preserve the so-called ring properties of the integers. We look at different ways to represent numbers, including the technique of continued fractions, which enjoys some surprising properties. Neither last, nor least, we construct numbers that turn traditional notions of arithmetic on its head, requiring us to reconsider even the definition of our beloved primes!Some aspects of this course will be experimental: we introduce you to a computer algebra system, Sage, and encourage you to infer patterns and solutions by experimentation.
Days of the Week:
Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday
- Level of Difficulty: All Levels
- Size: One-on-One
- Cost: Free
- Institution: Saylor
- Topics: Algebra, General Mathematics, Precalculus