What is number theory?
This is a gentle introduction to abstract mathematics, at a level where
majors and nonmajors alike can enjoy. It's not required by the major (although
it does fill the important second Algebra course requirement),
and doesn't serve as a prerequisite for lots of things to follow, but it's
an interesting survey of a branch of mathematics very, very different from
the standard (Calculus) fare.
By Number, read Integer. Number theory is the study of integers
only. So forget pi, e, and 1/2--you won't need them. This doesn't make
number theory easy. On the contrary, it makes it rather hard!
Here are some interesting questions which properly belong to number theory,
just to give you a sense of the field.
- Why are 3^7-3, 4^7-4, and 5^7-5 all divisible by 7?
- What integers a,b,c can be Pythagorean triples? (i.e. a^2+b^2=c^2)
- The factors of 28 (excluding 28 itself) are 1,2,4,7, and 14, which sum to 28.
6 also does this. Are there any odd numbers which do this?
- Is 2^127-1 prime? (Note that's a VERY large number. Trial division is not an option!)
- Is every even number the sum of two prime numbers?
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