On a small scale, the appearance of prime numbers seems random, but on a large scale there appears to be a pattern, which is still not fully understood. What are they? Why are the questions related to them so hard? One of the most interesting things about prime numbers is their distribution among the natural numbers. We explain what they are, why their study excites mathematicians and amateurs alike, and on the way we open a window to the mathematician’s world.įrom the beginning of human history, prime numbers aroused human curiosity. 'A composite could be divisible by other composites as well, but eventually, you can decompose it down to a set of prime numbers.' (An example: the number 48 has exactly two factors, 6 and 8, but you can break it down further into more than only two factors: 2 times 3 times 2 times 2 times 2.Prime numbers have attracted human attention from the early days of civilization. If a number is a composite number, in contrast, it's always divisible by some quantity of lower prime numbers. If I do that and add one to the product, that number has to be a prime.' So then I'll take all the prime numbers up to P and multiply them all together. 'Imagine that there is a last, biggest prime number. Proving that the list of prime numbers is infinite isn't that tough, Zegarelli says. So, with a composite number, you have multiple factor pairs, while with a prime number, you have only one factor pair, one times the number itself.' So, there are two ways to make six - 1 times 6, and 2 times 3. '2 times 3 equals 6, so 2 and 3 are factors of 6. (In ancient Greece, where the modern concept of infinity wasn't quite understood, Euclid described the quantity of prime numbers simply as 'more than any assigned multitude of prime numbers.')Īnother way of understanding prime numbers and composites is to think of them as the product of factors, Zegarelli says.

The Greek mathematician Euclid, back in circa 300 B.C.E., devised a Proof of the Infinitude of Primes, which may have been the first mathematical proof showing that there is an infinite number of prime numbers. There are plenty of other prime numbers - 2, 3, 7 and 11 also are on the list, and it keeps rolling from there. But the easier way to say that is to call it a prime number.' So, you could call 5 a non-rectangular number.

'The best you can do it is string it into a line, a single row of five coins. 'But if you take the number 5, no matter how you try, you can't put it into a rectangle,' Zegarelli notes. With the number 12, you could make it into more than one type of rectangle - you could have two rows of six coins, or three times four.' You can do that with eight, too, by putting four coins into two rows. You could form them into a rectangle, with two rows of three coins.

'Think about the number 6,' says Zegarelli, citing a composite number.

Give me a list of prime numbers to 100