# The Wiggins Prime Sieve:

The Wiggins Prime Sieve, or How Creating Math Creativity Sorts The Primes
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By Herb Wiggins, M.D.; Clinical Neurosciences; Discoverer/Creator of the Comparison Process/CP Theory/Model; 14 Mar. 2014
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Simplify, Simplfy, Simplify
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Efficiency, Efficiency, Efficiency
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Least energy, Least Energy, LE Rules
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This article is written with many goals. This article shows how mathematical creativity comes about, with protean consequences to solving NP not = P. It shows how Comparisons Process (CP) creates descriptive, basic information, and data, and how THAT information is sorted by Least Energy methods to create an efficient Prime sorter. This is the essence of the new complex system, cognitive neuroscience models, which uses nearly or at least far, far more universal processors, CP, and LE and the many Methods of Comparison (Dr. Paul B. Stark, UC Berkeley)  which create mathematical creativity. Step by step. This new model creates creativity without limit, and enlightens us as to much more of Where math creativity comes from, and HOW to create methods of prime sorting with higher and higher efficiencies, literally without limit. This article will make those who use the RSA for security, very, very nervous!!!
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Read on to find the wellsprings of math creativity (& indeed creativity of nearly ANY kind in any field) and why and HOW Thermodynamics drives information theory and HOW and why IT works. Then most will have found some of the water of math creativity, but NOT the well from which it comes.
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This is an efficient method to sieve out 75% of the Number line & which generates the Prime Multiples (PrM) & by exclusion/process of elimination the Primes.  These beginning paragraphs set the stage & background for understanding and the empirical methods which create empirical mathematics, and its applications.
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The basis of problem solving and creation of understanding is complex system, and thus very complicated. However, it’s manageable by the process of creating the simplicities which create/generate the complexities. We move from the simple to the complex, as a rule. Or in this case from the easily found primes to the primes with very long numbers of digits.
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For instance in understanding the complexity of cloud formations we must understand the three basic wave fronts of the weather. The Northerly dry, cooler fronts, the Southerly blowing warm, wet fronts, and these imposed on the prevailing Westerlies. We see these complex system stabilities, and then use those pattern recognitions to solve weather problems.
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Then we begin to see the repeating, complex interference patterns of the clouds as part and parcel of these 3 interfering, interacting waves fronts. We see bits and pieces of the waves, clearly from the West, the North and the South. And then we see the triangles of the cloud complex interference patterns. And thus can generally see the SW winds, the NW winds and those coming by combinations of the 3. From the simple fundamentals, we can develop an understanding of the complexity of the whole
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IN the same way the complex system Plate Tectonics is composed of 5 simple patterns, by pattern recogntiopns, to describe by words, but not mathematized/or mathematizable, the multiple complex systems of the earth’s surface of most all the plates. The plates, compared to each other, have these characteristics. There is sea floor spreading, and of various types, directions and so forth, but largely the Mid Atlantic spreading zones drive the westward movement of the Americas from the splitting off of the Americas from The Euro-Asian continents and Africa. So the American plated moves west and plows into the Pacific plate. That creates the next major fundamental pattern, subduction of the plates under the Western edges of the Americas. Which then creates the greatest quakes, and the volcanoes all along the Western Americas.
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Then there are the fault movements accompanying these complex movements, esp. shown by the San Andreas which is moving mostly northwards, carrying a chunk of the Pacific plate, the most basic element of Tectonics, the plates, northwards. There are many kinds of faults, a sort of hierarchy of them. Those are the great simple patternings which lead to understanding of the whole, but not completely.
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Then there is the hot spot which created the many basaltic lava flows in the Hawaiian isles, the Yellowstone hot spot from the Columbia River, & flows, etc.; the Deccan traps, and the Siberian traps. Thus the most of the complex plate interactions are described, but not easily predictably, by this model, which is descriptive & thus mathematically difficult to model. As Ulam stated Math must greatly advance in order to describe/model complex systems.
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The problem of finding, sorting to find the primes is just one of a long series of sorting problems which are unlimited in finding solutions to problems, a grand sort of NP not equal to P. If we can solve better and better this problem of creating Prime Multiples (PrM) which when compared to the number line which leaves the primes, & can necessarily exclude, & show the primes. We cannot Generate the primes. We can ONLY find them by excluding all numbers which are Prime Multiples. Those two exclude each other. A Prime cannot be a PrM, and the converse is also true. Therefore by finding a method which generates, creates and finds all possible PrM’s, then by exclusion, an efficient sorting process, we can find all the primes. & the consequences of that, are not just for math, alone. And this is yet another way to do it. An Eratosthenes Sieve, as modified by Paul Pritchard, and then greatly extended, sped up here.
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Now we go to Gauss’ Razor. Mathematical systems should ONLY be created for practical purposes such as in engineering and the sciences, etc. or for those which show us how mathematics works. The rest of it is so often fantasy, that Gauss rid those of inutility by his simple dictum.He sorted out the wheat from the chaff. & made math more efficient.
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The Razor is a form of empirical tester, sorter for mathematics. As we know Godel showed that math was incomplete. The Razor helps us complete that by testing, empirical sorting methods, which are least energy efficient by that sorting by T&E. That is what creates math by a creative finding, or creation of math to more precisely describe events. This has only begun, because complex systems cannot be describe very well, Mathematically.  As Ulam affirmed, math must greatly advance before it can describe complex systems.
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The empirical testing which Gauss’ Razor relies upon creates empirical methods to apply maths. It also understands that understanding maths are needed, too. And so he cut through the mass of math generations of too many useless types, to ONLY those which had a real, practical application, by sorting to show where those worked. As we know, the Lyapunov numbers model to some extent, but not completely (Godel), the least energy stabilities. Brain processes create “consciousness, as Friston’s paper in Aeon.com essays showed, very rigorously. That is a fine example of empirical, creative application of maths. Same is true of the S-curves of Whitehead, based upon what has been written before ( S-curves, least energy discussions).
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The rule applies to the not parallel line axiom which created the Not Euclidean geometries. But LOOK!  Spherical geometry is NOT Euclidean, because using the longitudes is not a Euclidean geometry. No parallel longitudinal lines!!! AND further, uses comparison of circumference to diameter & created THAT new method which solved the problems in spherical geometries, neatly. Information WAS added to create the new method, I.E. Pi. & by that example, the not Euclidean, 4D maths were created by at least 5 mathematicians, independently & used to model Einsteinian 4D space time. Again, pattern recognition. independently creating as in the Calculus, another useful form of mathematics.  Creation of Pi created the new math!!!
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The general rule is that our CNS works via Comparison processes to create recognitions, and from those create the pattern recognition by detecting the repeated events in existence created by complex systems. Ulam, Fermi and Pasta showed from the first, that the repeating, stable events in complex systems are those which allow us to understand, control and navigate those. The rule is Comparison Processing creates recognitions. Then pattern recognition shows the relationships among those repeating stabilities (Einstein, Physics and Reality, 1938). & then pattern recognition, pattern recognition and then Pattern recognition without many limits. Those create the efficient categories of Aristoteles, which create the highly efficient, least energy hierarchical arrangements of our knowledge in most all cases. From the simple CP to the complex. From the simple number line, to the simple repeating PrM’s, we can do the job.
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Therefore this new, efficient Prime sorter using Prime Multiples (PrM) to do the work, and it works, using basic math sieving, empirical means, PLUS the PrM to complete it.
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Essentially we must find, exclude the PrM’s to sort out the primes on the number lines. This must start with 1 and work up. First we note that every number ending in 2,  or is even, except for 2, is not prime. Then we must note the same for 3, and that creates the casting out method of 2, 3’s.  Then we note that 2X5 creates in all of its forms, those numbers which end in -0 and -5. Thus we can more quickly sieve out 60% of the PrM’s to give us the primes.
Then we note this further, efficient to use pattern: All primes numbers end in 1, 3, 7, 9. Thus we can begin sieve OUT by using this method, all Prime Multiples.
This creates the first set of primes, 1, 2, 3, 5, 7.  We then use those sorting rules to create the next Decade of primes, 10, ends in zero, not prime, but is PrM. 11, 13, 17, 19. And we cannot generate any of that quartet using a prime. Those are the remainders from the exclusion of the PrM’s. So we have then 21, 23, 27, 29. Using the cast out 3’s method efficient sieve, we are left with 23 and 29. Those are primes, because there are NO PrM’s generated by our tables which are those.
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Thus we have the original 3, 7, prime multiples, then more complete, 11, 13, 17, 19, primes to create the prime multiples (PrM’s). And then we enter those into the 20’s decade, & find 23, and 29.
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Then we enter those into the 31, 33, 37, 39, prime quartet in the 30’s decade, sieve out the 33 & 39 as 3’s by adding up the digits. Then we know we have 31 and 37. And using this same method without limit, in every decade and every centad (100 numbers), we can sieve out down to only 40 digits which is a 60% reduction in the search size. Using the prime 3 cast out system, by summing digits, efficiently, we can eliminate 13 to 14(most commonly) more of each quartet. Then we use the PrM generating tables to show which of that series are NOT prime, leaving by exclusion, sorting out and in a complementary pattern, the Primes.
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Same can be done by hand in about 15′ if efficient, the number line from 100 to 200, finding all the primes by this method too. But understanding that the squares of primes times all the primes is needed to further exclude all of the PrM. it uses no more, and rids about 5% of the remainder of possible primes. But because those numbers rise very quickly, there are not many of them, it’s simply that last bit of PrM elimination to get the primes.
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Then we carry out that method without limit, generating the Prime Multiples, which by comparing them to the quartets of each decade, 40 per 100, sorting, and excluding them from the number line, Thus leaving the primes.
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That’s how it’s done. See the next section of the PrM tables which do this, efficiently.
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Thus most all of the work is done and this empirical, efficient systems, works. It’s been checked through 2000 to show the ins and outs of it.
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Understand these basics. We Cannot generate the primes, but the pattern of generation of the PrM’s is the key to the pattern of the primes. We use the discovered primes to quickly generate a great many more primes, which keeps the system going and propels finding more of the primes. It’s accurate, can be done very efficiently by noting many simplifying rules to generate & ID all known PrM’s. and that’s the general method.  Thus the primes found, efficiently sort the number line to FIND the rest primes. Sounds like impossible bootstrapping, but empirically we are generating a lot more primes in each centad, than we need to find the primes about 4-10 centads ahead, and so forth. But we feed back in the primes sorted out by exclusion of PrM’s which then allows further work to be done. IN short, the Primes are used to sort out the primes.
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The Primes CANNOT be generated at all by arithmetic, but are exclusions and that is not an arithmetic rule, either, but is sorting, which is something quite different from simple mathematics. Using ONLY arithmetics  to find, sort out by exclusion the primes. Simply and effective, too.  See the prime multiples (PRM’s) tables below for examples of how to create those PrM’s, quickly.

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The below Prime Multiples shows us how to do this, using a new method, which is efficient, basic and can form the basis of much more mathematical understanding of primes and how to find them, without limits.
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Efficient methods are least energy methods. and ridding the number line of nearly 75% of the candidates for primes is efficient. As per the starting paragraphs, to show the way this can be created.
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So, in order to solve the prime number sorting problem, we must go to the basics. What is a prime number? It’s defined by the multiplication of 1 times a prime, which is the ONLY multiple a prime can have. It cannot be a multiple of any other number. Then we have the concept of the Prime Multiple. Which means the number is NOT prime but is composed of that set of primes multiplied against each other, the least of which are two primes. All of the other prime multiples are of the type of Prime times Prime, times Prime, etc. But must be at LEAST two prime multiples. Finding a single PrM does necessarily excludes the number from being prime, and we do NOT have to know any of the other multiples, either. That short cuts (efficiency) the problem, very quickly.
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No longer is it necessary to divide all numbers of the number line by at least 1/2 of that number and below, to find the primes. & it’s an Immense savings of a LOT of computing and time. Finding primes can be found using this mehtod very easily, if well organizing by only a calculator and limited by the number of digits that calculator can hold.  If the device can hold 13 digits, the primes can be generated up to the numbers in the trillions, and sequentially, and more besides.
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So if we generate PrM’s , then we know none of them are prime.
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Therefore PRIMES cannot be generated by any mathematical process in arithmetic, but are in fact, PRM exclusions and eliminations. A method of sorting, IOW.
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So by finding a prime multiple  greater than 2 and even, cannot be prime. Thus only the number line can create primes when it is generated. and ONLY by exclusion, elimination of the Prime Multiples  (PrM’s) can we find, sort, and detect the Primes. That basic, fundamental rule creates an efficient method, using only simple arithmetic, to generate all of the prime numbers, and exclude by a simple Comparison Process, using only what is prime.
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Thus we create this vast simplification of exclusion, elimination. Starting with the simple and moving up the number line to the complex.
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But what about 3? We can exclude that because it’s known that except for 1X3, which defines that prime, NO other number with a 3 as PrM can be prime. Thus the casting out of 3’s method. Sum up any sequences of number, such as 123, or 561, or etc. If the total is a number divisible by THREE, then that must be a prime multiplier of 3, and cannot be prime. This sorts out 1/3 of the number line and empirically 14/40 of the 1, 3, 7, 9, number ending quartets. Thus the 75% sorter.
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Empirically by sorting thru to 2000, we see that the numbers of primes can vary per centad, often from about 12, to as many as 18, or so. And with the higher #’s opf primes found, means the process is ever more efficient than 45% in those cases.
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and there is a very simple rule which can cast out 3’s which works better and better the larger then number of digits in the quartets. Another of unlimited efficiency rules which can be generated to speed up prime sorting out processes.
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So we can generate THESE decades, as we’ve largely excluded most all 3 multiples and the other five. It’s down from 10 numbers/decade to only 4. Vast simplification, as shown in the first paragraphs of this article. And when those 4 are also reduced by casting out 3 & exclusion, what’s left is about 25 numbers which are possible primes.
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Then we extend this method even further by the PrM Series starting with 3 as a demo, and simplification and then 7, 11, 13, etc., multiples of the other primes in sequence. That creates, generates a PrM & not a prime number, necessarily. And by extending the system using a lot of 3X3, 3X5, 3X 7, 3X9, 3X121, we see that NO number of the 3X higher primes in sequences CAN be prime. Thus we are left with 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, etc. as a prime series in the 10 decades of the first centad. By Exclusion/elimination of the PrM’s which cannot be primes.
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This simple sorting, exclusion, elimination method  cannot generate the primes, but excludes  PrM’s  and leaves primes by a complementary process. Primes Cannot be generated, but they can be found by the PrM exclusion process which generates all PrM’s!!!!
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The pattern of the primes is ONLY complementary to the complex interference patterns of the PrM’s. That’s the pattern which has been seen. & within which many other efficiency related patterns can be seen to speed up calculation. Without Limits!!!
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Pritchard used a method of 6, 2, 4 2, 6, 4, which only worked for a while, and which was ALSO generated by the series of PrM’s. And alswyas had to generate those. And did not r/o primes by casting out 3’s or any nubmer ending in 0, or 5. Thus there was a LOT more work to do.
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Therefore we have, if used accurately with good accounting, found a simple way to find the primes, by creating and ID’g All of the PrM’s in set sequnces, which necessarily gives us the primes.
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There is a simple pattern to creating those PrM series, which will now be shown. The pattern is IN the PrM’s which then creates a complementary pattern of primes. That’s the only pattern which can be seen as to how it arises. It will be shown below. There is NO prime sequence pattern.
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Part 2: The Prime sorting method using PrM tables.
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Below is the general arithmetic, simple method for creating the prime multiples (PrM’s, AKA composite numbers). This leaves out the numbers in the number line which are prime. Thus because NO prime can be generated, as generating numbers are created by arithmetic processes,. The method uses exclusion by  PrM’s.
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Note we do NOT have to fully factor any suspected prime, but we only have to show that it’s a 2 number, PrM. This greatly simplifies the process.
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Understanding this is easy. Any even number but for 2, (1X2), cannot be prime. That repeating of Four numbers is found in every decade of 1-10, 11-20, or 100-110, 111-120, etc. without limit. The end number of all prime numbers is always 1, 3, 7 & 9. That massively simplifies the sorting process to find the primes.  It’s an empirical form of mathematics, NOT logical.
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Also note that this table is relevant to understanding that quartet of end numbers:
end 1 can be generated by 1X 1; 3X 7; and 9X 9.
end 3 can only be created by 1X 3; and 7X 9.
end 7 by 1X 7; 3X 9.
& end 9, by 1X 9 and 3X 3, and 7X7;  such as 3X 13, or 23X 83 prime multiples.
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That vastly reduces the choices. And the above prime times prime pair of squares and cubes, etc., multiple prime creators finishes the job.
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also necessary to factor out every prime numbe squared. As those begin the process, it’s necessary to eliminate those when the number is p exp. x, >/= 2. For instance we know in the series 2400 to eliminate 343 & 2401, etc. as those are 7exp. 2 or greater.
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For example, we take the 3rd centad of the first 100 numbers.
31, 33, 37, 39  casting out 3’s in this series leaves only 31 and 37.
41, 43, 47, 49** casting out 3’s leave only 41, 43, 47 &49. The last is a prime multiple of 7X 7, which is a PrM, not prime. Those will be treated in detail at the end.
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These four end digits lists can be extended without limit to any number of digits, whether 3, 4, or 4000 or 4 Billions of digits.
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The last is more laborious, but using the above series of 3X any prime in series, and 7X  of each prime in series, and 11, 13, 17, 19, & so forth. THAT creates the multi primes which are end numbers of 1, 3, 7, and 9. And finishes the exacting job of excluding PrM’s, and leaves the numbers on the number line which remain that are primes.
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It’s not known if there is a casting out 7 method which works as well as casting out sums of digits divisible by 3 or not, but is being worked on. & likely can be found, altho this will only save about 5-6 numbers int he 40 number in each centad, but is a good extra example of the multiplicities of finding NEW, unexpected simplifying methods drive by brain pattern recognition.
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Thus we do NOT generate primes, and cannot. BUT, we can use the primes to generate PrM (prime multiples), which ARE not primes. and those can be All generated by the simple arithmetic series/ progression we see at first. Thus if the calculator has 9-12 digits we can calculate primes into the billions, but NOT billions of digits. That requires a computer.
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That is the key to sorting out primes. And as the primes are then sorted out 3-4 or more times faster than the PrM series can be generated, it never runs out of primes to test by the PrM method. For instance, the numbers of primes sorted out is just a fraction the number between each prime, such as 3, 7, 11, 13, 17 & 19 times 100. Thus the PrM generator is unlimited.
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This gives us 400 primes sorted out between the series of 7 and 111; and between 11 and 143, about 200 numbers sorted; and between 13, and 17, another 400+ numbers sorted. Thus a single pair of primes can create dozens of primes, and then the primes create the next series of PrM’s to be created. It’s self sustaining. .With some Positive feedback and the rising size of the factorials creates a very rapid growth of numbers of PrM created by this means, too.
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Here are the exemplary PrM series tables for each prime in the -1, -3, -7, -9 end number series.
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Three series
3X 1
3X  3 = 9
3X 5  = 15
Here is the start of the -1, -3, -7, -9 series
3X 7 = 21 cannot be prime
3X 9 = 27
3X11 = 33
3X 13 = 39 etc., and all sequences of the Three multiple prime series are NOT primes. This is the empirical legitimacy of the casting out of 3’s method.
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The next series starts the entire complicated system of creating by a simple subtraction, multiplication and adding, a general universal method not requiring ANY other math, to obtain the entire PrM listing, except for the Prime squares and their PrM’s, which will be shown to be simple at the end. As those are only about 3% of all of the numbers to be sieved, it completes the 97% of PrM created, which create by exclusion, sorting out of the primes.
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The simple mathematically clean and arithmetic progression series can be see. We do NOT need any longer to multiple ANY of the PrM’;s together, but uses a subtraction of the two primes in succession, a multiplication of the Prime series number in this case, 7, and  an addition to the previous PrM pair. And by casting out 9’s we can simply check if the multiplication is correct, or the addition is, too. These show the een number patterns of the Prime Multiples, which laid the basis for Pritchard’s work. altho he did NOT use the 1, 3, 7, 9, ending quartets simplification, either.
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7X 7 =     49 Not prime
7X  4=+28
7X 11 =   77
7X2 =   14
7X 13 =   91
7X4 =   28
7X 17 =  119
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Here we have done the 7 Series through 100.
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Now the next PrM series of 11
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11X 11 =  121
11X 2     +22
11X 13 =   143
11X4     +44
11X 17 =  187
11X2     +22
11X 19 =  201
11X 23
11X 29
11X 31
11X 37, etc.,
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Followed by 17X17, 19X19 and so forth covering each decad and each centad which is needed to be covered.
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Thus we can take say 2400 to 2500 and by finding the prime multiples from 2401 upwards to 2499, compute ALL the PrM’s for that centad and sort out the 2400 to 2500 Primes.
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With a large enough computer with enough registers and memory, we can compute ANY number of digits from say 1,000,500 to 1,000,600, of from 3,000,100,000 to 3,000,200,000.
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And we can generate PM’s at will over ANY range, once those primes have been found.
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Thus from ANY prime by a simple sorting process of the PrM’s which cover that range, we can create ANY listing of primes wherever we choose, as long as we can multiply correctly and do good accounting with the PrM’s which are simple and the Prime Squares, cubes, etc., too.
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There is thus NO limit to the number of digits we can process. Within polynomial time.
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Say a banking password uses 750-760 long prime digits. Those can be specifically created for a complete list of primes and be tested, easily and quickly using this method.
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And if the bank goes up to 2500 digits, and the computer is a RISC which can do these tasks specifically, then the numbers of primes can be found there, as well, and tested. Just get the password code in the chip for the primes, and go from there.
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& the banks can also create vast lists of primes, without doing ALL of the primes calculations but limiting those to the intervals of primes to be found.  The caveat is that the Primes which are prime multiples must be computed, too. But that’s doable. Only need a password chip or set of coded primes, to find out what they are.
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As the computers get more powerful and the registers get larger & larger, there is NO prime series which cannot be arithmetically done in polynomal time which cannot be found. Thus ALL primes within the limits of computational,l power, CAN be found. Without limits, too.
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Using his mthods, Pritchard found 22 sequential primes in the range of several millions of digits. And this method is even faster
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As each prime creates more PrM’s than the PrM exclude more PrM sorts more primes than it needs to create them. Thus we create the primes by using the PrM which are a self perpetuation series. This can be programmed, and solve the entire series of primes, too, within the limits desired. Simple, not complicated math other than a single subtraction, multiplying and addition, to sort out from the number line, by exclusion of PrM’s, of ALL the prime numbers in that target centad. Using a well designed RISC, to 1, list the number line, then establish the interval and the PrM’s which cover that interval, the entire prime numbers can be efficiently created, too.  This means every single number, within the limits of time and processing speed, no matter how long, can be found Within polynomial time to be Either PrM, Or Prime.
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Therefore, with a good enough computer it’s possible to VERY easily sort out all of the primes in polynomial time. The only limit is the power of the computer itself. And by using these methods, and clearing the memory stacks at intervals, and reprogramming the primes by the computer itself, the next series can be generated very quickly.
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And the only limits are the powers of the best RISC’s which can be made to sort out the primes by the above simply system. It simply uses the known primes and then goes efficiently from there.
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And it can create a series of primes starting at ANY spot, using the Known primes between those numbers too. It’s only a matter of sorting by arithmetic, now. Nothing more needs to be done. The numbers of squares are matters of using a good CRC, and using carefully chosen prime series, which will give all numbers between same 1,000,000, with the first PrM series, will give any prime series after that, as well. all that’s need are the data, to compute which PrM lie between say, 1,000,000 and 1,000,500. Then the rest of the prime lists both above and below can be computed at will, too. This can then generate LONG lists of any set of primes at any spot we wish to start at, once the computer has done the Prime sorting work to within the range of generating the PrM’s which lie around the Prime targets desired. In a practical sense ALL of the primes into the millions of digits are already known. & that allowed ever more primes to be found, from the prime lists which are a matter of public record, already.
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& that’s how it works. The technical delivery of the Primes is of course a lot of work, but then when most any prime number can be selected out and found, and then an entire list of primes can be generated to extend that list without limit, no prime number can escape from being found, regardless of how long it is. This is an encryption nightmare, which means as the computers become faster, more efficient by using better and more efficient methods without limit, that very likely, the race between prey and predator can be seen to extend to this. The better and better sorting methods which can be based on this method are unlimited as int he number line.
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And the time to find the primes is directly related to the power of the computer, regardless of digits. This method can do that. It’s just a matter of having a good accounting programmer.
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Further, within each Pr M series, there is an even larger pattern, than these & by finding those patterns, more efficiencies of computing the PrM’s can be found without limits, too.
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Thus we have shortened the time it takes to prove and find a prime, Empirically, substantially. & can further test the method for any possible flaws, OR correct mistakes found in the prime number list, which are likely to be substantial, the larger the digit lengths are. Which is experimental, empirical math at its best!!!
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And byh comparing the tables, we can immensely shorten the calculation times, because we have, like in the cases of sine, and cubes, and logs, we also have done the work. and the LTM efficiency of such tables, is ever more the case because it cuts down the sorting times, substantially. Thus the efficiency, robustness of the method and its sheer elegance, as most all new models must show. And fruitful without limit, besides, the last profoundly important element of any good model/theory, method. and that means, that LTM and archives of info are efficient, Least Energy methods, too. Which is why we use them to gain the LE saving, benefits & profits, above all!!!
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There is much more, but this shows how the method of creating Pr M quickly & robustly sorts & ID’s primes by simple exclusion, and when computer size properly and efficiently,can sort out primes to 10 mega digits in only minutes.
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This in short is the Wiggins Prime Multiples Sieve Method (my thanks to Paul Pritchard who proved an earlier form of what Eratosthenes created, first.) which robustly, empirically solves the order in which Primes exist, by showing the Not prime, Prime Multiples which sieves out efficiently, by complementary pattern of the NOT Prime multiples; thus sorting out which are primes. & Exactly.
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It’s the Surprise, unexpected use of the Primes to sort out and find the Primes, which is the key here. And as it’s an input, output system, can grow in power by a factor of 2, 4, 6 or more, the more work is done to create the PrM series, the vaster its power. and applicability.
As it’s unlimited, so the ability to solve the encryption problems with this method is unlimited. & nothing by this primes method can resist being broken, even by brute force. And because for each stage, new, more efficient sorting methods can be created, Without Limit, which increases the efficiencies of this method unlimited, then nothing is safe.
Primes are not therefore encryption proof but only so far as the computers cannot improve enough to brute force solve and find the primes.
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But it’s even worse than that. . As sorting is done best by QC, and calculation done best by chip computers, by combining the calculations of the one, by sorting using the QC, and very fast calculation & sroting can be done to find by sorting, NOT trial and error,m the real primes., Just don’t make a mistake and from time to time, check the multiplication results to make sure they fit.
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And moreover, there is yet another implication of this systems, and that is the very easy generating of MORE primes, once a prime succession has been found.
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And then similarly process the rest, and once we have the primes in those ranges, we can calculate out the rest of those numbers say between 2000 and 3000, thus sorting a new list of primes, where before we had a fewer, too.
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And this is hows it’s done and proven empirically and tested to be so.
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Thus we can in polynomial times, with powerful enough computers, break any 12-13 digits numbers or lower or higher, if we can get enough computer power and powerful enough algorithms, which can be generated without limits, by seeing the patterns without limits, and short cutting the problems of factorization. We can find a comparison process prime system of 2 adjacent Numbers of any primes within the range of the given digits size, which we can easily determine if prime  or not, and go from there.
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IOW, we can create  two lines of a single prime series and then generate all of the primes within a set limit such as 2000 to 3000, or 5 billions to 5 billions 1000., 5,000,000,1000. or any such limits, too.
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it’s simply a matter of generating all the prime multiples &  sorting to find the primes, and using the prime numbers thus found, to generate the “not prime” Prime multiples.
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QED.
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Keep It Simple!!!