American International Journal of Research in Formal, Applied & Natural Sciences Available online at http://www.iasir.net ISSN (Print): 2328-3777, ISSN (Online): 2328-3785, ISSN (CD-ROM): 2328-3793 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 -431605, 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 2.1.1 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 2.1.1 2 is prime For all values of k from 3 to n AIJRFANS 13-318; 2013, AIJRFANS All Rights Reserved Page 22
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 2.1.1 Time Taken in Seconds Sieve 1 Sieve 2.1.1 1 10 4 0.000006483 0.000000406 1 100 25 0.000074957 0.000009725 1 1000 168 0.005648124 0.000606951 1 10000 1229 0.097750231 0.044869363 1 100000 9592 7.102491985 3.611870137 As was the case with Sieve 1, for Sieve 2.1.1 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 2.1.1 Steps Taken Sieve 1 Sieve 2.1.1 Steps Difference 1 10 4 15 12 3 1 100 25 1133 628 505 1 1000 168 78022 39676 38346 1 10000 1229 5775223 2894496 2880727 1 100000 9592 455189150 227664778 227524372 One sees clearly that the number of required steps in the Sieve 2.1.1 has decreased to about half as compared to those in Sieve 1. III. Sieve 2 improvised to Sieve 2.1.2 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 2.1.2 : 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 2.1.2 is presented for comparative analysis : AIJRFANS 13-318; 2013, AIJRFANS All Rights Reserved Page 23
TABLE 3. RUNTIMES FOR SIEVE 2 AND SIEVE 2.1.2 Time Taken in Seconds Sieve 2 Sieve 2.1.2 1 10 4 0.000000405 0.000000405 1 100 25 0.000011344 0.000005672 1 1000 168 0.001494280 0.000316036 1 10000 1229 0.049147596 0.023151231 1 100000 9592 3.744000591 1.794596685 Time efficiency of Sieve 2.1.2 over Sieve 2 is remarkable. It shows that Sieve 2.1.2 is, in addition, better than both Sieve 1 and 2.1.1. 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 2.1.1. TABLE 4. STEPS TAKEN BY ALGORITHMS OF SIEVE 2 AND SIEVE 2.1.2 Steps Taken Sieve 2 Sieve 2.1.2 Steps Difference 1 10 4 9 10-1 1 100 25 616 376 240 1 1000 168 40043 20730 19313 1 10000 1229 2907640 1461014 1446626 1 100000 9592 227995678 114070446 113925232 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 2.1.3 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 2.1.3 : 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 2.1.3 Time Taken in Seconds Sieve 3 Sieve 2.1.3 1 10 4 0.000191647 0.000000000 1 100 25 0.000212716 0.000002431 1 1000 168 0.000625589 0.000045379 1 10000 1229 0.008435315 0.000961478 1 100000 9592 0.050981822 0.021859942 AIJRFANS 13-318; 2013, AIJRFANS All Rights Reserved Page 24
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 2.1.3. TABLE 6. STEPS TAKEN BY ALGORITHMS OF SIEVE 3 AND SIEVE 2.1.3 All numbers in the Steps Taken The Number of Primes Found Range Sieve 3 Sieve 2.1.3 Steps Difference 1 10 4 8 9-1 1 100 25 236 185 51 1 10000 1229 117527 65956 51571 1 100000 9592 2745694 1445440 1300254 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 2.1.1 Sieve 2.1.2 Sieve 2.1.3 1-10 4 0.000000406 0.000000405 0.000000000 1-100 25 0.000009725 0.000005672 0.000002431 1-1000 168 0.000606951 0.000316036 0.000045379 1-2000 303 0.002180646 0.001126789 0.000112233 1-3000 430 0.004652612 0.002529501 0.000196509 1-4000 550 0.007910210 0.004261214 0.000284432 1-5000 669 0.012171424 0.006451989 0.000384915 1-6000 783 0.017044045 0.009101421 0.000487424 1-7000 900 0.022949049 0.012071345 0.000597226 1-8000 1007 0.029222762 0.015332590 0.000713511 1-9000 1117 0.036491582 0.019006707 0.000831822 1-10000 1229 0.044869363 0.023151231 0.000961478 1-11000 1335 0.053508076 0.027485781 0.001094376 1-12000 1438 0.062820998 0.032159057 0.001245101 1-13000 1547 0.073423994 0.037374050 0.001382050 1-14000 1652 0.084922019 0.042804596 0.001526698 1-15000 1754 0.096463398 0.048522006 0.001674181 1-16000 1862 0.109628308 0.054934288 0.001826526 1-17000 1960 0.122295260 0.061126156 0.001974010 1-18000 2064 0.136520511 0.068241415 0.002133243 1-19000 2158 0.150309390 0.075017543 0.002286804 1-20000 2262 0.166257440 0.082934641 0.002454141 1-21000 2360 0.181996826 0.090756118 0.002617831 1-22000 2464 0.199535184 0.099519624 0.002784762 1-23000 2564 0.217172404 0.108275027 0.002954530 1-24000 2668 0.236337941 0.117636165 0.003130376 1-25000 2762 0.254445973 0.126887906 0.003305006 1-26000 2860 0.274131348 0.136645709 0.003483282 1-27000 2961 0.295353142 0.147104869 0.003669257 1-28000 3055 0.315779580 0.157217604 0.003850775 1-29000 3153 0.337784575 0.168203084 0.004038775 1-30000 3245 0.358802972 0.178815399 0.004223939 1-31000 3340 0.381867497 0.190211320 0.004412345 1-32000 3432 0.404776030 0.201569966 0.004603587 1-33000 3538 0.431990751 0.215020930 0.004806579 1-34000 3638 0.457380164 0.228336970 0.005006735 1-35000 3732 0.483675545 0.241161940 0.005207702 1-36000 3824 0.509570209 0.254077263 0.005407048 1-37000 3923 0.538130918 0.268263212 0.005613687 1-38000 4017 0.566065228 0.282102737 0.005821136 1-39000 4107 0.593608140 0.295722658 0.006024534 1-40000 4203 0.623614509 0.310550808 0.006235630 1-41000 4291 0.651863234 0.324805231 0.006441864 1-42000 4392 0.684456234 0.341333895 0.006662684 AIJRFANS 13-318; 2013, AIJRFANS All Rights Reserved Page 25
Numbers Range Number of Primes Sieve 2.1.1 Sieve 2.1.2 Sieve 2.1.3 1-43000 4494 0.718968543 0.358426562 0.006887556 1-44000 4579 0.748493971 0.373021737 0.007096221 1-45000 4675 0.782347061 0.389813359 0.007320688 1-46000 4761 0.813702659 0.405169856 0.007528542 1-47000 4851 0.846889644 0.421771857 0.007750982 1-48000 4946 0.882698914 0.439930130 0.007980715 1-49000 5035 0.917138696 0.457259823 0.008199914 1-50000 5133 0.956346565 0.476429006 0.008434915 1-51000 5222 0.992034283 0.494154959 0.008662623 1-52000 5319 1.031978758 0.513841549 0.008901270 1-53000 5408 1.069208973 0.531770900 0.009130598 1-54000 5500 1.108075685 0.551275162 0.009365599 1-55000 5590 1.147079346 0.570467035 0.009603031 1-56000 5683 1.187532720 0.590928318 0.009942567 1-57000 5782 1.231347821 0.612851063 0.010190939 1-58000 5873 1.272355472 0.633110974 0.010431207 1-59000 5963 1.313617167 0.653531739 0.010673095 1-60000 6057 1.358126736 0.675427742 0.010925924 1-61000 6145 1.400494556 0.696523527 0.011164977 1-62000 6232 1.443093325 0.717769226 0.011410108 1-63000 6320 1.486375623 0.739429418 0.011656454 1-64000 6413 1.533591754 0.762868324 0.011912929 1-65000 6493 1.574861958 0.783258295 0.012154818 1-66000 6591 1.626018000 0.808531423 0.012415750 1-67000 6675 1.670442077 0.830651893 0.012665742 1-68000 6774 1.724023896 0.856511308 0.012934777 1-69000 6854 1.767349547 0.878289000 0.013179097 1-70000 6935 1.812188117 0.900573160 0.013426253 1-71000 7033 1.867318105 0.927999382 0.013698530 1-72000 7128 1.920569302 0.954621333 0.013969591 1-73000 7218 1.973170194 0.979877444 0.014234574 1-74000 7301 2.021976227 1.004075240 0.014491049 1-75000 7393 2.076427143 1.031146935 0.014758869 1-76000 7484 2.128403663 1.058382320 0.015035602 1-77000 7567 2.178867263 1.083402619 0.015290861 1-78000 7662 2.237857343 1.112734373 0.015570431 1-79000 7746 2.290643400 1.138950746 0.015837035 1-80000 7837 2.347600317 1.167667042 0.016113363 1-81000 7925 2.404423932 1.195658887 0.016399821 1-82000 8017 2.463996652 1.225320048 0.016680607 1-83000 8106 2.521355501 1.254481224 0.016958151 1-84000 8190 2.577651580 1.282295198 0.017230833 1-85000 8277 2.636399771 1.312106273 0.017509998 1-86000 8362 2.694271976 1.341273527 0.017785111 1-87000 8450 2.753582143 1.371414820 0.018066302 1-88000 8543 2.817514939 1.403647620 0.018358027 1-89000 8619 2.871388077 1.430357089 0.018623010 1-90000 8713 2.939335742 1.463563117 0.018920002 1-91000 8802 3.003674117 1.495558485 0.019211727 1-92000 8887 3.065746355 1.526443272 0.019498995 1-93000 8984 3.144645465 1.562279284 0.019801660 1-94000 9070 3.209652783 1.594455360 0.020090144 1-95000 9157 3.275303516 1.627060920 0.020378627 1-96000 9252 3.347745370 1.663050493 0.020684128 1-97000 9336 3.411740968 1.695274379 0.020976258 1-98000 9418 3.475136503 1.727012867 0.021267983 1-99000 9505 3.542754761 1.760208360 0.021563760 1-100000 9592 3.611870137 1.794596685 0.021859942 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 13-318; 2013, AIJRFANS All Rights Reserved Page 26
FIGURE 1. RUNTIME COMPARISON OF THREE REFINED SIEVES References [1] David M. Burton, Elementary Number Theory, Tata McGraw-Hill Education, 2007. [2] Donald E. Knuth, The Art of Computer Programming, Volume 1: Fundamental Algorithms, Addison- Wesley, Reading, MA, 1968. [3] Neeraj Anant Pande, Evolution of Algorithms: A Case Study of Three Prime Generating Sieves, Journal of Science and Arts, 13, 3(24), 267-276, 2013. [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 7.3.1 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 13-318; 2013, AIJRFANS All Rights Reserved Page 27