What are Prime Numbers?

Prime numbers are an unusual set of infinite numbers, all of them whole (and not fractions or decimal), and all of them greater than one. When theories about prime numbers were first espoused, the number one was considered prime. However, in the modern sense, one can never be prime because it has only one divisor or factor, the number one. In today’s definition a prime number has exactly two divisors, the number one and the number itself.

The ancient Greeks created theories and development of the first sets of prime numbers, though there may have some Egyptian study into this matter too. What’s interesting is that the topic of primes wasn’t much touched or studied after the Ancient Greeks until well after the medieval period. Then, in the mid 17th century, mathematicians began to study primes with much greater focus, and this study continues today, with many methods evolved to find new primes.

In addition to finding prime numbers, mathematicians know that there are an infinite number, though they have not discovered all of them, and infinity suggests they cannot. Discovering the highest prime would be impossible. The best a mathematician could aim for is finding the highest known prime. Infinity means there would be another, and yet another in a never-ending sequence beyond what has been discovered.

The proof for the infinity of primes dates back to Euclid’s study on them. He developed a simple formula whereby two primes multiplied together plus the number one would sometimes or frequently reveal a new prime number. Euclid’s work didn’t always reveal new primes, even with small numbers. Here are working and non-working examples of Euclid’s formula:

2 X 3 = 6 +1 = 7 (a new prime)

5 X 7 = 35 +1= 36 (a number with numerous factors)

Other methods to evolve prime numbers in ancient times include using the Sieve of Eratosthenes, which was developed in approximately the third century BCE. In this method numbers are listed on a grid, and the grid can be fairly large. Each number viewed as a multiple of any number is crossed out until a person reaches the square roots of the highest number on the grid. These sieves could be large, and they are complicated to work with in comparison to how primes may be manipulated and found today. Today, because of the large numbers most people work with, computers are generally used to find new primes, and are much quicker at the job than people can be.

It still takes human effort to submit a possible prime number to many tests in order to assure it is prime, especially when it is extremely large. There are even prizes for finding new numbers which can be lucrative for mathematicians. Presently the largest known primes are over 10 million digits in length, but given the infinity of these special numbers it’s clear that someone is likely to break this threshold at a later point.