Many number puzzles involve prime numbers. Understanding the distribution of prime numbers is hence an important part of understanding the number system.

Generating Primes

A prime number is a number evenly divisible only by itself and by one. Seventeen is a prime number because there is no positive integer that can be divided into seventeen without a remainder except for one and 17. The smaller primes are well known: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, and so on. (Some mathematicians leave out 1.)

A straightforward method for generating primes was proposed by the Greek mathematician Eratosthenes (276 B.C.E.–194 B.C.E.). To begin this procedure, write down all the integers as high as you wish to go. Starting with 2, cross out every second number. Then, starting with 3, cross out every third number. Find the next uncrossed number, 5, and then cross out every fifth number. The next uncrossed number is 7, so cross...