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Prime numbers are integers greater than one with exactly two divisors. The ancient Greeks studied prime numbers, but interest waned until the mid-17th century. Mathematicians continue to find new primes using methods like Euclid’s formula and the sieve of Eratosthenes. Computers are now used to find large primes, but human effort is still required to verify them. The largest known prime numbers are over 10 million digits long, but there are likely to be larger ones in the infinite sequence.
Prime numbers are an unusual set of infinite numbers, all integers (and not fractions or decimals) and all greater than one. When prime number theories 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 developments of the first sets of prime numbers, although there may be some Egyptian studies on this topic. What’s interesting is that the subject of prime numbers was not much touched upon or studied after the ancient Greeks until well after the medieval period. Then, in the mid-17th century, mathematicians began to study prime numbers more closely, and this study continues today, with many methods evolved for finding new primes.
In addition to finding prime numbers, mathematicians know there are an infinite number of them, even if they haven’t discovered them all, and infinity suggests they can’t. Finding the highest prime number would be impossible. The best a mathematician can aspire to is to find the highest known prime number. Infinity means there would be one more, and one more in an infinite sequence beyond what has been discovered.
The proof of the infinity of prime numbers goes back to Euclid’s study of them. He developed a simple formula whereby two primes multiplied together plus the number one sometimes or frequently reveal a new prime. Euclid’s work didn’t always reveal new primes, even with small numbers. Here are some working and non-working examples of Euclid’s formula:
2 X 3 = 6 +1 = 7 (a new prime number)
5 X 7 = 35 +1= 36 (a number with many factors)
Other methods of evolving prime numbers in ancient times include the use of the sieve of Eratosthenes, which was developed in approximately the 3rd century BC. In this method the numbers are listed on a grid and the grid can be quite large. Any number seen 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 are complicated to use compared to how prime numbers can be manipulated and found today. Today, due to the large numbers most people work with, computers are generally used to find new prime numbers and are much faster at the job than people are.
It still takes human effort to put a possible prime number through a lot of tests to make sure it’s prime, especially when it’s extremely large. There are also prizes for finding new numbers that can be profitable for mathematicians. Currently the largest known prime numbers are over 10 million digits long, but given the infinity of these special numbers it is clear that someone is likely to exceed this threshold at a later date.
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