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# Efficient prime test

, 12 Nov 2013 CPOL
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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.

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 First Prev Next
 This is not "efficient" prime test fulloflove28-Feb-14 17:15 fulloflove 28-Feb-14 17:15
 Not the prime example of optimization CaldasGSM12-Nov-13 7:37 CaldasGSM 12-Nov-13 7:37
 Re: Not the prime example of optimization arnaldo.skywalker12-Nov-13 8:36 arnaldo.skywalker 12-Nov-13 8:36
 Message Removed bharat_h0312-Nov-13 22:49 bharat_h03 12-Nov-13 22:49
 Re: Not the prime example of optimization arnaldo.skywalker14-Nov-13 4:07 arnaldo.skywalker 14-Nov-13 4:07
 Re: Not the prime example of optimization bharat_h0321-Nov-13 0:01 bharat_h03 21-Nov-13 0:01
 There are several prime sieves. Pete O'Hanlon12-Nov-13 5:59 Pete O'Hanlon 12-Nov-13 5:59
 Re: There are several prime sieves. arnaldo.skywalker12-Nov-13 6:53 arnaldo.skywalker 12-Nov-13 6:53
 Re: There are several prime sieves. PIEBALDconsult12-Nov-13 11:06 PIEBALDconsult 12-Nov-13 11:06
 Re: There are several prime sieves. arnaldo.skywalker14-Nov-13 4:11 arnaldo.skywalker 14-Nov-13 4:11
 Yes but you have to decide whether a number is prime that has a cost.
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