What does a prime number consists in?

T he most common definition is that “a number is prime if it is divided by 1 and by itself” or ”it is every number having just two divisors, this is, itself and the unit”. Thus, for instance, number 7 is a prime number since it is divided just by 1 and 7. On the contrary, number 6 is not a prime number because it can be divided by 1, 2, 3 and 6.

Concerning number 1, is it a prime one?

Having in mind what was previously said, the answer is “No”. Actually, number 1 only has one divisor.

And number 0, is it a prime one?

According to the same previous definition, the answer keeps on being “No”. Since a prime number can be divided by itself and zero cannot be divided by zero, because `0/0` is an indeterminacy.

And as far as negative numbers are concerned? Are they prime numbers?

This is a question involving more complexity since the previous definition just works with positive whole number. In order to include the negative numbers we would have to change the definition as follows: “A prime number is a whole which exactly admits four divisors”. Accordingly, the only divisors of number -5 are {-5, -1, 1, 5}. Therefore, the number -5 is also a prime number.

The answer is not consensual and it is not significance as well. The study of prime numbers had been carried out long before the emergence of negative numbers. And the truth is that when the negative numbers appeared, the mathematicians did not want to change all the current theorems so as to include the negative numbers, and so, it was agreed that when we talk about prime numbers we mean the positive whole numbers superior to one.

What does it mean to factorize a number?

Let us remember that all the prime numbers have two divisors: the number itself and the unit. All the remaining numbers are called compound numbers and they have, at least, 3 divisors. Every compound number can be figured by the product of several prime numbers. Number 60, for instance, can be written as follows: `60=2xx2xx3xx5=2^2xx3xx5`, This is the process through which a number is factorized and from which results one of the most important mathematical laws known as the Fundamental Theorem of Arithmetic, according to which “every natural number superior to one can either be a prime one or it can be written as a product of prime numbers.”

Do we know all the prime numbers?

There is not any pattern able to find the prime numbers. However, more than 2000 years ago Euclid proved that there is an infinite amount of prime numbers. Nowadays, it is already possible to find prime numbers containing thousands and thousands of digits thanks to the help of super computers that are able to perform lots and lots of calculations.

What is a sieve of Eratosthenes?

Eratosthenes was a Greek mathematician whose reputation is due to the fact that he found a method able to find prime numbers. Let's us suppose the existence of a table containing the first 1000 natural numbers. The first step is to point out the first prime number of the table which is number 2. The following step is to erase (from which comes the notion of sieve) all the multiples of that number. Then comes the third number of the table (number 3), which hasn't been erased yet and then we will eliminate all the multiples of number 3. Then the following number comes in the table and whose multiples will be eliminated too and we keep on doing the same thing till we reach the last number of the table. Therefore, it will be easy to find out all the prime numbers between 1 and 1000. After this process of “cleaning”, the table will include 168 numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991 e 997.

Check out our List of Questions to get to know a little more about the most diverse topics related to mathematics. If you have any pertinent (math) question whose answer can not easily be found, send us an email on the Contact page with the question. We will be happy to respond. In the event that you detect any errors in our answers, do not hesitate to contact us!