Keywords: Algorithm, Sieve, Prime Number Mathematics Subject Classification (MSC) 2010 : 11Y11, 11Y16, 65Y04, 65Y20, 68Q25

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1 American International Journal of Research in Formal, Applied & Natural Sciences Available online at ISSN (Print): , ISSN (Online): , ISSN (CD-ROM): AIJRFANS is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research) Algorithms of Three Prime Generating Sieves Improvised by Skipping Even Divisors (Except 2) Neeraj Anant Pande Department of Mathematics & Statistics, Yeshwant Mahavidyalaya (College), Nanded , INDIA Abstract: Three elementary classical prime generating sieves are well known. In this paper, a property of positive integers is employed that if any of them is not divisible by 2, then it is also not divisible by any even number and hence there remains no necessity to examine any even divisors during primality tests. This significantly improves all the three classical sieves and gives a new generation of three better sieves. The corresponding algorithms when implemented on electronic computer show very noteworthy difference given by this simple property. Keywords: Algorithm, Sieve, Prime Number Mathematics Subject Classification (MSC) 2010 : 11Y11, 11Y16, 65Y04, 65Y20, 68Q25 I. Introduction Prime numbers still stand as a mystery for mathematicians [1]. There are many conjectures about prime numbers which are yet unresolved. Definition (Prime Number or Prime) : An integer p > 1 is prime number if, and only if, it only divisors are ±1 and ±p. The lack of a simple precise formula for prime numbers has made all the work about primes quite analytic. Huge databases of primes are generated and analysed in detail. So regarding the primes, the very basic requirement many times happens to be availability of these databases. For generation of such databases, methods called sieves are employed which are implemented on electronic computers in programming languages based on algorithms [2]. II. Sieve 1 improvised to Sieve Using mere fundamental property that positive divisors of a positive integer k cannot range beyond 1 to k itself, of which if we omit 1 and k,, test range for the non-trivial divisor remains 2 to k 1, following Sieve 1 was discussed by author in [3] : For all values of k from 2 to n For values of integer d from 2 to k 1 If checks do not stop for any value of d till k 1, k is prime As it happens to be, this sieve was admitted to be very inefficient at the very original source of its appearance. In fact, for the same reason it was followed by its refinements as Sieve 2 and Sieve 3 in the same discussion, which we will revisit in the coming sections. There is another scope for increasing efficiency of this sieve by appealing to a property of divisibility of positive integers that if 2 doesn t divide k, then 2j doesn t divide k, i.e., if a number is not even then it doesn t have any even divisors. This leads to following Sieve is prime For all values of k from 3 to n AIJRFANS ; 2013, AIJRFANS All Rights Reserved Page 22

2 For values of integer d from 2 to k 1 and only odd values after 2 If checks do not stop for any value of d till k 1, k is prime Runtimes requirements both sieves are as follows : TABLE 1. RUNTIMES FOR SIEVE 1 AND SIEVE Time Taken in Seconds Sieve 1 Sieve As was the case with Sieve 1, for Sieve also the time grows rapidly for higher ranges relative to lower ranges, but the comparative time requirement is quite less as compared to original Sieve 1. It is almost reduced to half as the number of checks of the divisors has reduced almost to half after omitting all even values beyond 2. [3] has done time requirement calculation. If, in addition, here step requirements are measured, the following table gives the quantum of the improvement over Sieve 1 of [3]. TABLE 2. STEPS TAKEN BY ALGORITHMS OF SIEVE 1 AND SIEVE Steps Taken Sieve 1 Sieve Steps Difference One sees clearly that the number of required steps in the Sieve has decreased to about half as compared to those in Sieve 1. III. Sieve 2 improvised to Sieve Along the same lines of improvement of Sieve 1 to Sieve 2.1.1, there is equal scope of improvement of Sieve 2 in [3] to a newer version. Sieve 2 of [3] adopted the following approach : For all values of k from 2 to n For values of integer d from 2 to k/2 If checks do not stop for any value of d till k/2, k is prime This was a clear enhancement of Sieve 1 s method as number of divisors was reduced to half owing to the property of positive integers that none of their divisors can exceed half of themselves. To this better made form, if new property of skipping testing of even divisors after 2 is applied we get a further refined version, what we call Sieve : 2 is prime For all values of k from 3 to n For values of integer d from 2 to k/2 and only odd values after 2 If checks do not stop for any value of d till k/2, k is prime A table of runtime requirements of Sieve 2 and Sieve is presented for comparative analysis : AIJRFANS ; 2013, AIJRFANS All Rights Reserved Page 23

3 TABLE 3. RUNTIMES FOR SIEVE 2 AND SIEVE Time Taken in Seconds Sieve 2 Sieve Time efficiency of Sieve over Sieve 2 is remarkable. It shows that Sieve is, in addition, better than both Sieve 1 and Now if number of checks required for primality tests are to be compared in terms of steps taken by both sieve 2 and 2.1.2, we see a similar difference between them as between those of Sieve 1 and Sieve TABLE 4. STEPS TAKEN BY ALGORITHMS OF SIEVE 2 AND SIEVE Steps Taken Sieve 2 Sieve Steps Difference Here too the steps are about halved. The reason is very same that during all primality tests, half of the divisor values which are even except 2 are omitted during checks and that leaves half checks redundant, directly avoiding their execution. The data is drawn for all numbers up to 10 5 and will be used to plot these values. IV. Sieve 3 improvised to Sieve The best sieve amongst all presented in [3] is Sieve 3. It considers precise range for divisors, namely, from 2 to square root of a number for search of possible divisors : For all values of k from 2 to n For values of integer d from 2 to k If checks do not stop for any value of d till k, k is prime Finally, introduction of similar refinement of neglecting even numbers larger than 2 for test as divisors for this sieve also leads to the best version of the sieve amongst all being presented in this paper, enumerated as Sieve : 2 is prime For all values of k from 3 to n For values of integer d from 2 to k and only odd values after 2 If checks do not stop for any value of d till k, k is prime It is found that the process of refinement continues. The chart of time requirements shows similar trend of less time consumption for this version. TABLE 5. RUNTIMES FOR SIEVE 3 AND SIEVE Time Taken in Seconds Sieve 3 Sieve AIJRFANS ; 2013, AIJRFANS All Rights Reserved Page 24

4 The time for first range of numbers 1-10 is so small and less than 1 nanosecond that it is seen to be just 0 in the order of magnitude of nanosecond. It is actually not zero but only very small. Now what follows is the table describing the number of steps taken by Sieve 3 and its improved form Sieve TABLE 6. STEPS TAKEN BY ALGORITHMS OF SIEVE 3 AND SIEVE All numbers in the Steps Taken The Number of Primes Found Range Sieve 3 Sieve Steps Difference Same trend continues in this case also as the number of steps becomes reduced to nearly half. V. Comparative Analysis Of the 3 previous algorithms, each one is refined in a similar manner to yield 3 more. Each time newly obtained algorithm has proven to be better than all the previous. Originally presented algorithms for Sieves 1, 2 and 3 were analysed for performance comparison in [3]. Here an exhaustive comparison for newly presented versions becomes due. The next table shows all related generated data about time requirements of newly introduced sieves. TABLE 7. RUNTIMES IN SECONDS FOR THE THREE REFINED SIEVES Numbers Range Number of Primes Sieve Sieve Sieve AIJRFANS ; 2013, AIJRFANS All Rights Reserved Page 25

5 Numbers Range Number of Primes Sieve Sieve Sieve These readings are not to be considered proportationate to those given in [3], the reason being those values are obtained by execution of algorithms on a different electronic computer and these are on a different machine with advanced hardware configuration. One more point needs be made clear that these execution times are obtained at the smallest possible nanolevel and any programming language does not guarantee 100% precesion or accuracy at that scale. Repeated exceutions of the same algorithms on same machine are prone to give differenet time-readings. So the readings presented here are intended to serve as an instantneous example. A graphically representation of this data follows : AIJRFANS ; 2013, AIJRFANS All Rights Reserved Page 26

6 FIGURE 1. RUNTIME COMPARISON OF THREE REFINED SIEVES References [1] David M. Burton, Elementary Number Theory, Tata McGraw-Hill Education, [2] Donald E. Knuth, The Art of Computer Programming, Volume 1: Fundamental Algorithms, Addison- Wesley, Reading, MA, [3] Neeraj Anant Pande, Evolution of Algorithms: A Case Study of Three Prime Generating Sieves, Journal of Science and Arts, 13, 3(24), , [4] Herbert Schildt, Java : The Complete Reference, 7 th Edition, Tata McGraw - Hill Education, 2006 Acknowledgments Author owes sincere thanks to Honourable Prof. Shreehari for helping him revise this paper by accurately determining the number of steps required during primality tests by bringing it to notice that each time when determination begins by dividing by 2 it needs be counted as a step, which was missed in the first draft of the paper. The author expresses his heartfelt thanks to the Java (7 Update 25) Programming Language Development Team and the NetBeans IDE Development Team whose software were used (and also because these are available free to use) in implementing the algorithms on their platforms. Thanks will also be due to University Grants Commission (U.G.C.), New Delhi of the Government of India for funding awaited for this work under a proposed Research Project. AIJRFANS ; 2013, AIJRFANS All Rights Reserved Page 27