Number Theory is a branch of mathematics that goes back to the Pythagoreans in the 5th century B.C. that studies the properties of the whole numbers.
The field of Number Theory, as with many fields in mathematics, began as an abstract exercise and eventually found countless applications in the modern world. This page has links to a variety of sites that explore both sides of Number Theory.
In the Advanced Topics class, we connect a variety of topics under the umbrella of Number Theory.
The Partition Number [p(n)] for a given value represents the number of ways that the number can be written as a sum of smaller natural numbers (so 0 is not included).
For example: 4 = 4 = 3+1 = 2+2 = 2+1+1=1+1+1+1, so p(4)=5 (5 different ways to partition the number 4)
The Palindrome Order of a number is found by adding the reverse digits of the number until a palindrome is reached. The number of additions necessary is the palindrome order.
What would it mean to have a palindrome order of 0?
Can you find a number that has a palindrome order of 4?
Does there exist a number for which the palindrome order is infinite?
Sizes of Infinity:
A countably infinite set implies that the list of numbers can be aligned in a one-to-one correspondence with the natural numbers, and so is countable
An uncountably infinite set implies that the list of numbers cannot be aligned in a one-to-one correspondence with the natural numbers
The whole numbers, perfect squares, integers, rational numbers and primes, are examples of countably infinite sets
The real numbers, points on a circle, and points on a line segment represent uncountably infinite sets
Mersenne Numbers are numbers in the form 2p-1 (where p is a prime). Mersenne Primes are Mersenne numbers that have been found to be prime. By the 1950's, there were 18 known Mersenne primes, each under 1000 digits long (the first ones were discovered in the 1400s and 1500s). It would seem there would be hundreds of these things known by now, but it turns out that, currently (as of 1/10/05), there are 41 Mersenne Primes known to the human race and the 41st is over 7 million digits long.
Perfect Numbers are numbers whose proper factors add to the number itself. The first perfect number is 6 because 1+2+3=6 and 1,2, and 3 are all the proper factors of 6. There are currently 41 known perfect numbers. Coincidence? No. If a Mersenne prime is in the form 2p-1, if you multiply 2p-1 by 2p-1, you get a perfect number!
A discussion of Mersenne Primes (which are incredibly huge and don't have applications in the real world) naturally leads to a discussion of the RSA encryption system and its use of very large prime numbers.
The University of Illinois Mathematics Department changed their stamp meter after finding this Mersenne prime in 1963
Our discussion of encryption leads to learning how digital steganography works -- but to understand that one must cover, number systems, bit, bytes, pixels, image storage format, ASCII, etc. Once we have discussed systems that use digital technology and the binary system, we are free to learn about 1D and 2D barcoding systems and the mathematics behind error detection and correction.
Polygonal Numbers (pronounced with an emphasis on the "y", not the "gon") are visually engaging and the analysis of them leads to a myriad of conjectures and discoveries of patterns...
Fermat's Engima, by Simon Singh and Fermat's Last Theorem, by Amir Aczel
both excellent overviews of the history and solution to Fermat's Last Theorem
Curious & Interesting Numbers, by David Wells
this book is an amazing attempt to describe every number that is interesting in one way or another - it is in numerical order - reading it will give you a great overview of the field of Number Theory
