Proof, a movie starring Anthony Hopkins, Gwyneth Paltrow, Jake Gyllenhaal, and Hope Davis, is about a math proof. After a retired mathematician dies, a notebook is found in his locked desk drawer. In the notebook is one of the most important proofs in the history of mathematics. This particular proof solves a problem in number theory that had, until then, never been solved. This post isn't about the movie or the proof; it's about number theory.
What is number theory? Simply put, it's the study of integers. Integers include all positive numbers, their negatives, and zero. They don't include fractions and decimals. One might ask how can mathematicians study just numbers. What's there to know about them? I mean, you use them to count and to order. What more is there? Well, a more pointed question might be: What more can you do with them? Some basic properties of numbers are:
You can add them: a+b=c
You can subtract them (subtracting is adding): a+(-b)=d
You can multiply them: a*b=e
You can divide them (dividing is multiplying): a*(1/b)=f
OK, well, we learned how to do all that stuff in elementary school. Why, then, is there a branch of mathematics about it? Well, consider the following:
The Pythagoreans, an ancient sect of mathematicians, said that a number is perfect if it equals the sum of its positive divisors, excluding itself. For example, 1+2+3=6 and 1+2+4+7+14+28. But 10 is not perfect because 1+2+5=8 not 10 and 12 is not perfect because 1+2+3+4+6=16 not 12. Euclid was able to create a formula that would find all even perfect numbers. But are there any odd perfect numbers? We know there aren't any between 0 and 10^300, but mathematicians have yet to prove that there exist no odd perfect numbers.
What about this?
Are there infinitely many primes p such that p+2 is also prime? For example, 3 is prime which means it is only divisible by itself and 1. 3+2 equals 5, which is also prime. Conversely, 13 is prime, but 13+2 is not. And 97 is prime, but 99 is not. In 1966, Chen Jingrun showed that there are infinitely many primes p such that p+2 is the product of, at most, two primes. So if 13 is prime, which it is, then 15 must be the product of two primes (in this case, 5 and 3).
Proofs usually have two parts
A proof that something exists.
A proof that nothing else exists.
Without these two crucial parts, a theorem is not complete, and, therefore, worthless. In the words of the greatest contributor to mathematics in history, Carl F. Gauss, "(1/2)*proof=0, and it is demanded for proof that every doubt becomes impossible."
There are many more questions we can ask about the properties of numbers. What are their relationship to one another; are there patterns in operations involving numbers; how do they act when under certain conditions. And, though this may seem meaningless to the average person, these questions are important because they bring us closer to understanding where we are, what we are, and who we are. These patterns, rules, and properties belong to the universe we inhabit. Just as we toddled around our home when we were infants, so we continue in our quest to explore, to know, and to understand our larger home.