Which numbers are prime

This means 1 and 7 are the only factors of 7. So, 7 is a prime number because it could not be divided into groups of equal numbers. Definition of a Prime Number: Any whole number greater than 1 that is divisible only by 1 and itself, is defined as a prime number. Prime numbers created human curiosity since ancient times. Even today, mathematicians are trying to find prime numbers with mystical properties. Euclid proposed the theorem on prime numbers - There are infinitely many prime numbers.

Do you know all the prime numbers from 1 to ? Did you check if each number is divisible by the smaller numbers? Then, you definitely invested a lot of time and effort. Eratosthenes was one of the greatest scientists, who lived a few decades after Euclid, designed a smart way to determine all the prime numbers up to a given number.

This method is called the Sieve of Eratosthenes. Suppose you have to find the prime numbers up to n, we will generate the list of all numbers from 2 to n. Similarly, assign the next value of p which is a prime number greater than 2. There are 25 prime numbers from 1 to The complete list of prime numbers from 1 to is given below:.

Check out few more interesting articles related to prime numbers for better understanding. There is a difference between prime numbers and co-prime numbers. Co-prime numbers are always considered in pairs, while a single number can be interpreted as a prime number. If a pair of numbers has no common factor apart from 1, then the numbers are called co-prime numbers.

Co-prime numbers can be prime or composite, the only criteria to be met is that the GCF of co-prime numbers is always 1. This formula will give you all the prime numbers greater than Let's substitute a few whole numbers and check. A prime number chart is a chart that shows the list of prime numbers in a systematic order. Given below is the prime number chart from numbers 1 to that shows the list of odd prime numbers highlighted in yellow.

The set of prime numbers between any two numbers can be found by following a pattern. The following figure shows a few prime numbers encircled and striking off all the numbers divisible by these prime numbers.

You can follow this pattern until you reach the square root of the larger number, that is, The number 15 has more than two factors: 1, 3, 5, and Hence, it is a composite number. On the other hand, 13 has just two factors: 1 and Hence, it is a prime number. Therefore, 13 is a prime number. The factors of 20 are 1, 2, 4, 5, 10, and Thus, 20 has more than two factors. Since the number of factors of 20 is more than two numbers, it is NOT a prime number 20 is a composite number.

To check whether is a prime number or a composite we need to identify its factors first. The factors of are 1, 11, , Thus, has more than two factors. Since the number of factors of is more than two numbers, it is NOT a prime number. Hence, is a composite number. Numbers having only two factors i.

A whole number that can be written as the product of two smaller numbers is called a composite number. A number that cannot be broken down in this way is called a prime number. The numbers. In fact, these are the first 10 prime numbers you can check this yourself, if you wish! Looking at this short list of prime numbers can already reveal a few interesting observations.

First, except for the number 2, all prime numbers are odd, since an even number is divisible by 2, which makes it composite. So, the distance between any two prime numbers in a row called successive prime numbers is at least 2. In our list, we find successive prime numbers whose difference is exactly 2 such as the pairs 3,5 and 17, There are also larger gaps between successive prime numbers, like the six-number gap between 23 and 29; each of the numbers 24, 25, 26, 27, and 28 is a composite number.

Another interesting observation is that in each of the first and second groups of 10 numbers meaning between 1—10 and 11—20 there are four prime numbers, but in the third group of 10 21—30 there are only two. What does this mean?

Do prime numbers become rarer as the numbers grow? Can anyone promise us that we will be able to keep finding more and more prime numbers indefinitely? Do not continue reading! Write all the numbers up to and mark the prime numbers. Check how many pairs with a difference of two are there. Check how many prime numbers there are in each group of 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. From 31 through 40, there are again only 2 primes: 31 and From 91 through , there is only one prime: That even seems to make sense; as numbers get bigger, there are more little building blocks from which they might be made.

Do the primes ever stop? Suppose for a moment that they do eventually stop. Dividing by any of those primes would result in a remainder of 1. So, either q is prime itself and certainly greater than p or it is divisible by some prime we have not yet listed which, therefore, must also be greater than p. Either way, the assumption that there is a greatest prime — p was supposedly our greatest prime number — leads to a contradiction!

Prime Numbers. Prime Numbers Topics: Mathematical Language. Meaning An informal sense Building numbers from smaller building blocks: Any counting number, other than 1, can be built by adding two or more smaller counting numbers.

A formal definition A prime number is a positive integer that has exactly two distinct whole number factors or divisors , namely 1 and the number itself. Clarifying two common confusions Two common confusions: The number 1 is not prime.

The number 2 is prime. It is the only even prime. The number 1 is not prime. Why not?