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Efficient prime test

, 12 Nov 2013
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The IsPrime algorithm will always find a number that divides n if it is composed.

Introduction

Prime numbers are essential in many fields of mathematics and computer science, especially cryptography. An interesting problem arises when we need to decide whether an integer is a prime number.

The straightforward, naïve algorithm for deciding if a number n is prime follows a procedure in which a loop from 2 to n-1 checks every time whether the number representing the step of the loop divides n and in that case returns false.

The previous algorithm runs in O (n). An improvement to the running time of the Naïve prime test can be achieved if we ask ourselves the question: Is it really necessary to loop from 2 to n-1? The answer to this question is no, is not necessary, it would be enough to loop only from 2 to squart(n).

Correctness of the IsPrime algorithm: Let’s assume that all numbers that divide n are greater than squart(n). If this is the case then the smallest number that can divide n is squart(n)+1, but, if n is composed then the smallest numbers dividing n must be greater or equal than [squart(n)+1] [squart(n)+1], but this product is greater than n, which is a contradiction, though there must be at least a number dividing n less than or equal to squart(n) proving that the IsPrime algorithm will always find a number that divides n if it is composed.

License

This article, along with any associated source code and files, is licensed under The Code Project Open License (CPOL)

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About the Author

arnaldo.skywalker
Software Developer
Cuba Cuba
I'm a programmer and mathematician, graduated of Computer Science. Lover of Jazz, music, cinema and art in general.

Comments and Discussions

 
QuestionThis is not "efficient" prime test Pinmemberfulloflove28-Feb-14 17:15 
QuestionNot the prime example of optimization PinmemberCaldasGSM12-Nov-13 7:37 
AnswerRe: Not the prime example of optimization Pinmemberarnaldo.skywalker12-Nov-13 8:36 
OK, the key point of the tip is the idea of going until sqrt(n), It's not intended to be the most efficient prime test in the world, though, thanks for the suggestion, is true that improvement could be made.
GeneralMessage Removed Pinmemberbharat_h0312-Nov-13 22:49 
GeneralRe: Not the prime example of optimization Pinmemberarnaldo.skywalker14-Nov-13 4:07 
GeneralRe: Not the prime example of optimization Pinmemberbharat_h0321-Nov-13 0:01 
QuestionThere are several prime sieves. PinprotectorPete O'Hanlon12-Nov-13 5:59 
AnswerRe: There are several prime sieves. Pinmemberarnaldo.skywalker12-Nov-13 6:53 
GeneralRe: There are several prime sieves. PinprofessionalPIEBALDconsult12-Nov-13 11:06 
GeneralRe: There are several prime sieves. Pinmemberarnaldo.skywalker14-Nov-13 4:11 

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