Why is 11 a prime number

As happens so often, my initial neat and tidy answer for why things are the way they are ended up being only part of the story. Thanks to my friend for asking the question and helping me learn more about the messy history of primality.

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See Subscription Options. Go Paperless with Digital. In the positive whole numbers, each prime number p has two properties: The number p cannot be written as the product of two whole numbers, neither of which is a unit. Get smart. Sign up for our email newsletter.

Sign Up. Read More Previous. Support science journalism. Historically, the sieve of Eratosthenes dating from the Greek mathematics implements this technique in a relatively efficient manner.

More modern techniques include the sieve of Atkin, probabilistic algorithms, and the cyclotomic AKS test. Is this number a prime number? As a consequence, 11 is only a multiple of 1 and Find out more: What is a prime number?

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. We will demonstrate the sieve of Eratosthenes on the list of prime numbers smaller than , which is hopefully still in front of you Figure 2. Circle the number 2, since it is the first prime number, and then erase all its higher multiples, namely all the composite even numbers.

Move on to the next non-erased number, the number 3. Since it was not erased, it is not a product of smaller numbers, and we can circle it knowing that it is prime. Again, erase all its higher multiples. Notice that some of them, such as 6, have been already deleted, while others, such as 9, will be erased now. The next non-erased number—5—will be circled. Again, erase all its higher multiples: 10, 15, and 20 have already been deleted, but 25 and 35, for instance, should be erased now.

Continue in the same manner. Until when? All numbers smaller than that were not erased are prime numbers and can be safely circled! What is the frequency of prime numbers?

How many prime numbers are there, approximately, between 1,, and 1,, one million and one million plus one thousand and how many between 1,,, and 1,,, one billion and one billion plus one thousand? Can we estimate the number of prime numbers between one trillion 1,,,, and one trillion plus one thousand? Calculations reveal that prime numbers become more and more rare as numbers get larger. But is it possible to state an accurate theorem that will express exactly how rare they are?

Such a theorem was first stated as a conjecture by the great mathematician Carl Friedrich Gauss in , at the age of The nineteenth-century mathematician Bernhard Riemann Figure 1 , who influenced the study of prime numbers in modern times more than anyone else, developed further tools needed to deal with it. But a formal proof of the theorem was given only in , a century after it had been stated. It is interesting to note that both men were born around the time of the death of Riemann.

The precise formulation of the prime number theorem, even more so the details of its proof, require advanced mathematics that we cannot discuss here. But put less precisely, the prime number theorem states that the frequency of prime numbers around x is inversely proportional to the number of digits in x. Indeed, computer calculations show that there are 75 prime numbers in the first window, 49 in the second and only 37 in the third, between one trillion and one trillion plus one thousand.

The same information can be pictured as a graph, shown below Figure 3. Yes Is 11 a Composite Number? No Multiples of 11 11, 22, 33, 44, 55, 66, 77, 88, 99, Is 11 an Odd Number? Yes Is 11 an Even Number? No Is 11 a Perfect Square? No Is 11 a Perfect Cube? No Cube Root of 11 2.