What are the two numbers? She adds the two numbers together. Her answer is Write all the possible pairs of prime numbers Emma could be thinking of. He multiplies it by 10 and then rounds it to the nearest hundred. His answer is Write all the possible prime numbers Chen could have chosen. Registration for the free maths resources is quick, easy and available for all staff at your school.
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One to one interventions that transform maths attainment. Find out more. Support for your school next term Personalised one to one maths lessons designed to plug gaps, build confidence and boost progress Register your interest. Group Created with Sketch. Register for FREE now. What Is A Prime Number?
Ellie Williams. What is a prime number? A prime number is a number greater than 1 with only two factors — themselves and 1. A prime number cannot be divided by any other numbers without leaving a remainder. Prime number examples How to work out if a number is a prime number or not. What are the prime numbers? There are 8 prime numbers under 2, 3, 5, 7, 11, 13, 17 and The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 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.
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.
If you do, it can't be a prime number. If you don't get a whole number, next try dividing it by prime numbers: 3, 5, 7, 11 9 is divisible by 3 and so on, always dividing by a prime number see table below. History Government U. Cities U. Updated February 21, Factmonster Staff. Some facts: The only even prime number is 2. All other even numbers can be divided by 2.
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