Imagine Magazine - Johns Hopkins - March/April 2011 - (Page 14)

prime, pat by James Pommersheim, PhD 2,3,4,5... What in mathematics could be simpler than the natural numbers? There are no ugly decimals, scary fractions, or untidy square roots, just the plain old numbers that we use every day for counting. Known as number theory, the study of the natural numbers is one of the oldest and most revered branches of mathematics. Number theory has fascinated the greatest mathematical thinkers throughout history. In fact, Carl Friedrich Gauss, who revolutionized many areas of mathematics and physics in the early 19th century, called number theory “the Queen of mathematics.” Prime Questions To see the appeal of number theory, one need only consider the prime numbers. Take the numbers 30 and 31. They might seem pretty much the same—but not to a number theorist! The number 30 has many factors—it is divisible by 1, 2, 3, 5, 6, 10, 15, and 30. On the other hand, 31 is a prime number, being divisible only by 1 and 31. The prime numbers are the building blocks of the natural numbers. Every number can be factored into prime numbers. How many prime numbers are there? If you have the feeling that there is no end to the list of prime numbers, you are right. But in mathematics, you cannot know for sure that the statement is correct until you prove the statement. To prove mathematically that there are infinitely many primes, we may need a little help from Euclid, the famous ancient Greek geometer and author of the famous text called Elements, written in roughly 300 BCE. Euclid was by no means immune to the charms of number theory. In fact, 3 of the 13 books of the Elements were devoted to number theory. Here is Euclid’s idea. Suppose there were only a finite number of primes. For illustration, let’s imagine that the primes end at 19, so that a complete list of primes is: 2, 3, 5, 7, 11, 13, 17, and 19. Our proof will be complete if we can somehow produce a prime larger than any prime in our list, i.e., larger than 19. To pull this off, Euclid does something wildly clever. He multiplies all the primes together and adds 1. In our example, we arrive at a grand total of 2*3*5*7*11*13*17*19 + 1 = 9,699,691. Now it’s possible that this number is prime. In that case, we’re very happy because we’ve produced a new prime that wasn’t on our original list of primes. But what if 9,699,691 is not prime? Notice that 9,699,691 cannot be divisible by 2 since it is one more than a multiple of 2. Similarly, 9,699,691 cannot be divisible by 3, 5, 7, 11, 13, 17, or 19. So the prime factors of 9,699,691 must be bigger than 19. And voilà, we’ve produced a prime bigger than 19. In this manner, no matter how many primes we’ve found, we can always find a bigger one. The prime numbers do indeed go on forever. But is there a pattern to the prime numbers? Sometimes a pair of primes, like 11 and 13, are only two numbers apart. Such primes are called twin primes. As another example, 1,019 and 1,021 are twin primes. In the spirit of Euclid’s proof, one might wonder if there are infinitely many twin primes. Many mathematicians suspect there are, but no one has been able to give a proof. This is just one of many questions about prime numbers that is easy to ask, but that has remained unsolved over the millennia. If you make a list of prime numbers, you’ll notice that they tend to spread out the further out you go. That is, as you consider larger numbers, you will tend to find that there are fewer primes on average. Of the numbers less than 100, 25 percent are prime; but of the numbers less than 1,000,000, only about 8 percent are prime. What is the chance that a really big randomly chosen number is prime? It depends on how big the number is. Take a 10-sided die with the digits 0, 1 ,…, 9 on it, and roll it 13 times to produce a 13-digit number. You’ve now chosen a random number between 0 and 10 trillion. I just did that at my desk and rolled 4,659,571,442,361. What is the probability that the number you obtain is a prime number? There is a formula that expresses this probability in terms of logarithms. Conjectured 14 imagine Mar/Apr 2011

Table of Contents for the Digital Edition of Imagine Magazine - Johns Hopkins - March/April 2011

Imagine Magazine - Johns Hopkins - March/April 2011
Contents
Big Picture
In My Own Words
Problem Solving
Orange County Math Circle
Number Theory
Count Me In
National Mathematics Competitions
Math at the Science Fair
Just My Speed
MathPath
When Origami Meets Rocket Science
Selected Opportunities & Resources
Making a Difference
High School Options for Gifted Students
Off the Shelf
Word Wise
Exploring Career Options
One Step Ahead
Planning Ahead for College
Students Review
Mark Your Calendar
Knossos Games

Imagine Magazine - Johns Hopkins - March/April 2011