If a number is a composite, in contrast, it's always divisible by some quantity of lower prime numbers. So why have primes held such fascination among mathematicians for thousands of years? As Zegarelli explains, a lot of higher mathematics is based upon primes.
But there's also cryptography, in which primes have a critical importance, because really large numbers possess a particularly valuable characteristic.
There's no quick, easy way to tell if they're prime or composite, he says. The difficulty of discerning between huge primes and huge composites makes it possible for a cryptographer to come up with huge composite numbers that are factors of two really big primes, composed of hundreds of digits.
If I've got one of those factors in my pocket, I've got the key to the house. That's why mathematicians have continued to labor to come up with increasingly bigger primes, in an ongoing project called the Great Internet Mersenne Prime Search.
In , that project led to the discovery of a prime number that consisted of 23,, digits, enough to fill 9, book pages, as University of Portsmouth England mathematician Ittay Weiss described it in The Conversation.
It took 14 years of computations to come up with the gigantic prime, which is more than , times bigger than the estimated number of atoms in the observable universe! Though many have believed that primes are random, in a paper, two Stanford University mathematicians described a previously unknown apparent pattern, in which primes tended to be followed by other primes ending in certain digits, as this Wired article details.
For example, among the first billion prime numbers, a prime ending in 9 is about 65 percent more likely to be followed by a prime ending in one than it is to be followed by a prime ending in nine.
Sign up for our Newsletter! Mobile Newsletter banner close. Mobile Newsletter chat close. Mobile Newsletter chat dots. Mobile Newsletter chat avatar. If you looking for primes then, half of all possible numbers can be taken off the table right away the evens , along with all multiples of three, four, five, and so on. It might seem that this would leave no numbers after a certain point, but in fact, we know that there are an infinite number of primes — though they do become less frequent as we go on.
An extremely complex mathematical proof can assure you that combinations of prime numbers can be multiplied to produce any number at all — though if you can understand that proof, this article, frankly, is not for you.
In a sense, we can define primes according to this status as a basic-level number: primes are the total set of numbers which are left over when we rewrite all numbers as their lowest possible combination of integers. When no further factoring can be done, all numbers left over are primes. This is why primes are so relevant in certain fields — primes have very special properties for factorization. You can imagine then how unfathomably hard it might be then to factor a number 50 or even digits long.
While that may sound like a problem, for the uses of prime numbers it is actually an opportunity. Modern encryption algorithms exploit the fact that we can easily take two large primes and multiply them together to get a new, super-large number, but that no computer yet created can take that super-large number and quickly figure out which two primes went into making it.
A modern super-computer could chew on a bit factorization problem for longer than the current age of the universe, and still not get the answer. And that means we are constantly using prime numbers, and relying on their odd numerical properties for protection of the cyber-age way of life. Can you find any patterns? Or does the list of prime numbers up to seem random to you?
Prime numbers have occupied human attention since ancient times and were even associated with the supernatural. Even today, in modern times, there are people trying to provide prime numbers with mystical properties. The idea that signals based on prime numbers could serve as a basis for communication with extraterrestrial cultures continues to ignite the imagination of many people to this day.
It is commonly assumed that serious interest in prime numbers started in the days of Pythagoras. Pythagoras was an ancient Greek mathematician. His students, the Pythagoreans—partly scientists and partly mystics—lived in the sixth century BC. They did not leave written evidence and what we know about them comes from stories that were passed down orally.
Three hundred years later, in the third century BC, Alexandria in modern Egypt was the cultural capital of the Greek world. Euclid Figure 1 , who lived in Alexandria in the days of Ptolemy the first, may be known to you from Euclidean geometry, which is named after him. Euclidean geometry has been taught in schools for more than 2, years. But Euclid was also interested in numbers.
This is a good place to say a few words about the concepts of theorem and mathematical proof. A theorem is a statement that is expressed in a mathematical language and can be said with certainty to be either valid or invalid. To be more precise, this theorem claims that if we write a finite list of prime numbers, we will always be able to find another prime number that is not on the list.
To prove this theorem, it is not enough to point out an additional prime number for a specific given list. For instance, if we point out 31 as a prime number outside the list of first 10 primes mentioned before, we will indeed show that that list did not include all prime numbers. But perhaps by adding 31 we have now found all of the prime numbers, and there are no more? What we need to do, and what Euclid did 2, years ago, is to present a convincing argument why, for any finite list, as long as it may be, we can find a prime number that is not included in it.
If you pick a number that is not composite, then that number is prime itself. Otherwise, you can write the number you chose as a product of two smaller numbers. If each of the smaller numbers is prime, you have expressed your number as a product of prime numbers. If not, write the smaller composite numbers as products of still smaller numbers, and so forth. In this process, you keep replacing any of the composite numbers with products of smaller numbers.
Since it is impossible to do this forever, this process must end and all the smaller numbers you end up with can no longer be broken down, meaning they are prime numbers. As an example, let us break down the number 72 into its prime factors:. We will demonstrate the idea using the list of the first 10 primes but notice that this same idea works for any finite list of prime numbers. Let us multiply all the numbers in the list and add one to the result.
Let us give the name N to the number we get. The value of N does not actually matter since the argument should be valid for any list. The number N , just like any other natural number, can be written as a product of prime numbers. Who are these primes, the prime factors of N?
We do not know, because we have not calculated them, but there is one thing we know for sure: they all divide N. But the number N leaves a remainder of one when divided by any of the prime numbers on our list 2, 3, 5, 7,…, 23, This is supposed to be a complete list of our primes, but none of them divides N. So, the prime factors of N are not on that list and, in particular, there must be new prime numbers beyond Have you found all the prime numbers smaller than ? Which method did you use?
Did you check each number individually, to see if it is divisible by smaller numbers? If this is the way you chose, you definitely invested a lot of time. Eratosthenes Figure 1 , one of the greatest scholars of the Hellenistic period, lived a few decades after Euclid. He served as the chief librarian in the library of Alexandria , the first library in history and the biggest in the ancient world.
Among other things, he designed a clever way to find all the prime numbers up to a given number. Since this method is based on the idea of sieving sifting the composite numbers, it is called the Sieve of Eratosthenes.
0コメント