Prime numbers are numbers that are divisible by itself and 1 only or the numbers whose only factors are the number itself and

1. Sometimes a question that arises after reading the definition of prime numbers is what factors are. A number is a factor of another number if it can divide it perfectly without any remainder. A few prime numbers, for example, are 2, 3, 5, 7, 11, 13, 17 etc.

The numbers that have more than two factors are called composite numbers. One is neither prime nor composite because it has only one unique factor, which is itself.

There is no largest prime number as for every prime number p there exists a prime number which is greater than p. Euclid’s proof that there are infinitely many prime numbers is the most famous and accepted proof. The largest known prime number has more than 23 million digits giving it one million more digits than the previous record holder.

**Determination of prime numbers**:

Since there is no largest prime number, finding a new prime number by using the most powerful of the supercomputers is also not possible. There are a few algorithms that are generated for the sole purpose of finding out if a given natural number is prime or not. One of the easiest of such algorithms has been explained below.

Let n be the number we need to find out about. We are not aware if n is a prime number or a composite number. The first step is to take the square root of the number n. The second step comprises of rounding this number to the next highest whole number and name it as m. After finding m, the following few steps are to be performed, in which we find the quotients q_{2}, q_{3}, … q_{m}.

q_{m} = n / m

q_{(m-1)} = n / (m-1)

q_{(m-2)} = n / (m-2)

q_{(m-3)} = n / (m-3)

. . .

q_{3} = n / 3

q_{2} = n / 2

q

q

q

q

. . .

q

q

The number n is said to be prime only and only if none of the q’s derived are whole numbers.

The reason why prime numbers and finding which number is prime is so important that they are a big help in encrypting data. The main use for prime numbers now has become encryption. Any big hacker can decrypt data or find out that the numbers are prime or not, but among the unlimited number of prime numbers we have, it is almost impossible to decrypt the data that has been encrypted using prime numbers. The fact that we do not even know the largest prime number available makes prime numbers extremely special.

A Mersenne prime number should be reducible to the form 2

Fermat prime is a Fermat number that is also a prime number. The Fermat number F_{n} is of form 2^{m}+1, where m = 2^{n} where n is an integer.

These few quick tricks will help the student identify if a number is prime or not quickly without any loss of precious time. The students can instead use this time to solve the much harder questions.

As a practice the students can try and find all the prime numbers till 1000, using the method mentioned above. The table for all prime numbers up to 1000 shall look like this:

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

941 947 953 967 971 977 983 991 997

Another method that was designed to find prime numbers is as follows. This method is called the Sieve of Eratosthenes. Eratosthenes was a Greek mathematician who designed this simple way to find prime numbers up to 100.

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

941 947 953 967 971 977 983 991 997

Another method that was designed to find prime numbers is as follows. This method is called the Sieve of Eratosthenes. Eratosthenes was a Greek mathematician who designed this simple way to find prime numbers up to 100.

This is a long yet easy way of finding the prime number up to 100. We will not need to find out prime numbers up to 100 in the examination so no need to think and worry about these methods for long. As long as the student is able to identify if the given number is prime or not it will be enough from an exam point of view.

Factoring of composite numbers is also a part of the concept of prime numbers that the students may find interesting to know.

Factoring is dividing a number into parts such that the numbers obtained shall multiply to give the original number. To factor into prime numbers means factoring a given number such that the numbers obtained are prime numbers:

Factoring is dividing a number into parts such that the numbers obtained shall multiply to give the original number. To factor into prime numbers means factoring a given number such that the numbers obtained are prime numbers:

Ex: Factor 36 into prime numbers.

Dividing 36 by 2,

36/2 = 18

Now keep dividing the result by prime number till we reach 1.

18/2 = 9

9/3 = 3

3/3 = 1.

Thus the prime factors for the number 36 are 2 x 2 x 3 x 3.

Below there are a few questions that may help the students check their understanding and knowledge about prime numbers.

Possible Answers:

The above-given questions are from past SAT question papers. Students are advised to solve more questions like this to get a proper idea about the concept of prime numbers.