The "nuggets" from which arithmetic is made, prime numbers are numbers that are divisible only by themselves and the number 1 (for example, 23 is prime because it's only divisible by 23 and 1, whereas 24 is "composite" because it can be factored in various ways-8 times 3, for example, or 2 times 12). Sounds dull, right? Well, there's a lot more to the primes than their divisibility, as you'll discover by reading this quick summary.
Instructions
1. There are an infinite number of primes. Everyone is familiar with small primes like 5, 11, and 13, but since ancient times mathematicians have known that there is no such thing as a "largest" prime. As a result, computer scientists are constantly discovering larger and larger primes, consisting of hundreds (or even thousands, or millions) of digits.
2. There may (or may not) be an infinite amount of prime "twins." A prime twin consists of two primes that are separated by 2: for example, 11 and 13 are prime twins, as are 17 and 19. Computer scientists have identified prime twins with hundreds of digits each, but to date, it's unknown whether there are an infinite number of prime twins, the way there are an infinite number of primes.
3. Prime numbers are essential for cryptography. When you multiply two primes together, the result is a number with two prime factors (besides itself and 1, of course). When you multiply two huge primes together (say, consisting of 100 or 200 digits each), the result is a number so large that it would be practically impossible to factor without knowing one of the primes involved. This is what most cryptography systems are based on: basically, a message can only be decoded if you know one of these huge primes.
4. No one knows the exact location of large prime numbers. Unless it's laboriously calculated, it's impossible to know the exact numerical location of a prime number, a fact that has driven mathematicians crazy for centuries. All that can be stated with certainty is that the distribution of primes follows an approximate formula-meaning there will be about X primes below a given number.
5. The most important prime number theorem is the "Riemann Hypothesis." You'd need an advanced postdoctoral degree in mathematics to understand it, but the Riemann Hypothesis is an important statement about the distribution of prime numbers in the upper reaches of the mathematical stratosphere. Currently the most important unsolved problem in mathematics, if the Riemann Hypothesis were somehow proven false, this would invalidate many of the mathematical results of the last 100 years!
Tags: prime twins, infinite number, prime numbers, Riemann Hypothesis, digits each