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- 1. Probabilistic approach to prime counting Chris De Corte chrisdecorte@yahoo.com KAIZY BVBA Beekveldstraat 22 9300 Aalst, Belgium February 2, 2014 1
- 2. The goal of this document is to share with the mathematical community the ﬁnding of a new mathematical relationship with regard to prime counting that originated out of our probabilistic approach to the occurrence of primes. One of the side results will be a new formula for the natural logarithmic function as well. 2
- 3. CONTENTS CONTENTS Contents 1 Key-Words 4 2 Introduction 4 3 Derivation of formula’s 4 3.1 Lemma: Probability of ﬁnding a prime . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Theorem: Probabilistic prime counting formula . . . . . . . . . . . . . . . . . 6 3.3 Corollary: new formula for natural logarithm . . . . . . . . . . . . . . . . . . 7 4 Method and techniques 7 4.1 Probabilistic Prime Counting formula . . . . . . . . . . . . . . . . . . . . . . 7 4.2 New formula for natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . 8 5 Discussions 10 6 Acknowledgments 10 7 References 11 3
- 4. 3 1 DERIVATION OF FORMULA’S Key-Words Prime counting, Logarithmic Integral, Riemann, Gauss, Euler-Mascheroni. 2 Introduction The following document originated out of our interest for primes. We remain surprised by the fact that the young Riemann was at the origin of such a complex formula as a proposition for the prime-counting function [1] π(x) = x ln(x) − 1 (1) or others [1] using the logarithmic integral: x li(x) = 2 dt ln(t) (2) Nowhere did we ever see a theoretical explanation where this ﬁrst formula came from. For this reason, we started our quest to derive a (better) formula based on theoretical grounds. The question was: how? 3 3.1 Derivation of formula’s Lemma: Probability of ﬁnding a prime The probability (Pr) to ﬁnd a next prime immediately after an existing prime pi can be represented by: x=pi (1 − 1/pi ) P r(pi+1 ) = (3) i=2 Proof: Suppose that we have a classical x-axis which starts at 0 and which has nodes that count up according to the natural numbers. We will see every natural number as a node on this x-axis. 4
- 5. 3.1 Lemma: Probability of ﬁnding a prime 3 DERIVATION OF FORMULA’S Suppose our task is to touch every natural number with a graph. The easiest way to do this would be to create a sinus like function with a period T=2. This graph would immediately go through 0, 1, 2, ... hence through all the natural numbers. We call it the 1-graph. There is no way to ﬁnd a vacant spot any more on this x-axis now. So, if a vacant spot would represent a chance of a possible prime number, there would be no chance of ﬁnding a prime number any longer. The above is the same as saying that every natural number is divisible by 1. We will now delete our 1-graph and start with the new challenge to touch all the nodes from 2 up on the x-axis using a combination of diﬀerent graphs than the 1-graph. Remember: our challenge is to touch all the nodes on the x-axis without using the 1-graph. So, we will try to fulﬁll the fundamental theorem of arithmetic [1] using pi − graph s only. So, immediately after 1 but before 2: what is our chance of ﬁnding a prime? This will be 100% since all the nodes are still free and the next number that we will ﬁnd will only be divisible by 1 or itself. This is because there are no natural numbers in between. The next number we ﬁnd is 2 and 2 is a prime. The most logic ﬁrst graph in this new challenge will be the 2-graph and it will go through: 0, 2, 4, ... The probability of ﬁnding vacant spots on our x-axis (from 2 onward) will now be represented by the formula (1 − 1/2). So, the number of vacant nodes along the x-axis immediately diminishes with 1/2. These vacant spots can temporarily be interpreted as possible prime candidates over a short distance along the x-axis. And it seems to be true as the next vacant spot (prime) is node 3. The next logical graph that we will add will be the 3-graph and will go through 0, 3, 6, ... One would at ﬁrst think that the number of possible vacancies along the x-axis would immediately diminish with an extra 1/3 but that is not completely correct since certain points (like 6, 12, 18, ...) would be touched by the 2-graph as well as by the 3-graph and we should not double count them. The correct number of vacant spots on our x-axis (from 3 onward) will now be represented by the formula (1 − 1/2).(1 − 1/3). From 3 onward the chance of ﬁnding a next prime will hence be 33%. If we continue with the same reasoning, the next vacant spot (prime) we ﬁnd is node 5. So, the next logical graph will have to go through 0, 5, 10, ... Now, we must watch out not to overdo with avoiding the double counting (1/2x3, 1/2x5 and 1/3x5) hence we will have to correct for a triple counting with adding a factor for 1/(2x3x5). The correct number of vacant spots (from 5 onward) on our x-axis will however simplify to the formula (1−1/2).(1−1/3).(1−1/5). Our new formula will be valid till we ﬁnd the next vacancy: prime 7. We are now ready to believe that the probability for ﬁnding a vacancy (read a prime) immediately after x = pi will be represented by: x=pi (1 − 1/pi ) P r(pi+1 ) = (4) i=2 5
- 6. 3.2 Theorem: Probabilistic prime counting formula 3 DERIVATION OF FORMULA’S In the tables, we call this probability the total probability. The individual contribution to this probability by each diﬀerent prime pi , we will call the individual probability and is simply calculated as (1 − 1/pi ). We can mention already that the contribution of this total probability over a distance ∆x to the prime counting will then be represented by: x=pi (1 − 1/pi ) ∆x. (5) i=2 3.2 Theorem: Probabilistic prime counting formula The probabilistic prime counting formula can be represented as follows: x x=pi (1 − 1/pi ).dx with α ≈ 1.7810292 π(x = pi ) = α. 2 (6) i=2 α can be very closely approximated as: α = eγ where γ ≈ 0.57721 is the Euler − M ascheroniconstant (7) Proof (including an heuristic argument) In order to derive the probabilistic prime counting formula, we have to multiply each new probability of ﬁnding a prime with the distance over which this probability is valid (until the next prime) and sum up all these values. This can be approximated as: x=pi π(x = pi ) = ∆(x) x=pi {[ {P r(pi+1 ).∆(x)} = (1 − 1/pi )].(pi − pi−1 )} (8) i=2 i=2 This is like calculating a surface over an irregular step function and can be represented with an integral symbol. This method is an integration by parts. However after initial tests on the ﬁrst 50 million primes (see Method and Techniques), We came to the conclusion that the above probabilistic prime counting function should be corrected with a constant α. Therefore, if we refer to table 1, we can check that α = 50847534/28549683. As a conclusion, we come to formula 6. 6
- 7. 3.3 3.3 Corollary: new formula for natural logarithm 4 METHOD AND TECHNIQUES Corollary: new formula for natural logarithm We can now compare formula 2 with formula 6 and conclude the following new formula for the natural logarithm: ln(x = pi ) = β x=pi i=2 (1 − 1/pi ) with β = 0.5614575 (9) β can be very closely approximated as: β = e−γ where γ ≈ 0.57721 is the Euler − M ascheroniconstant (10) Proof The proof is trivial if we explain that we introduce β = 1/α. We must mention however that we have optimized β in this case to come as close to li(x) as possible (using a slightly diﬀerent α. Therefore, if we refer to table 1, we can check that β = 28549683/50849229. 4 Method and techniques We have developed a C++ program to verify the above logic for the ﬁrst 50.7 million primes. The result is stored in 1000 separate ﬁles of each around 7 Megabytes. A summary of this information can be found in tables 1 and 2. 4.1 Probabilistic Prime Counting formula We will now explain the calculation steps done and refer to the respective column in both tables where this information can be found. In the ﬁrst column of both ﬁle types we have put a counter that basically will count the primes. In the second column, we have listed the ﬁrst 50.7 million primes. In a third calculation we have calculated the individual probability of the prime of that row using the formula: Ind P rob(pi ) = (1 − 1/pi ) (11) Because of space reasons, we have not listed this calculation in table 1 but it can be found in the third column of the separate ﬁles. The calculation is however trivial. 7
- 8. 4.2 New formula for natural logarithm 4 METHOD AND TECHNIQUES In a next calculation, we calculated what we call the total probability of all the primes until that prime according to the formula: x=pi (1 − 1/pi ) T ot P rob(pi ) = (12) i=2 The result of this calculation can be found in column 3 of table 1 and in column 4 of the separate ﬁles. In the next column (My Pure), we calculated formula 6 but with α = 1. So, it comes to calculating formula 7. The reason for this is that we are then able to calculate the results for all the primes ﬁrst and then estimate α afterwards based on the last prime calculated. We multiplied the previous probabilistic value with the diﬀerence of this prime with the previous prime (∆x) and added this value to all the previous values (integration by parts). In the next column (My Integr), we did exactly the same as in the previous column but we corrected α = 1.7810292 from prime 2 onward. In the next column, we simulated the logarithmic integral (formula 2) in a similar discrete fashion: li(pi ) = li(pi−1 ) + pi − pi−1 ln(pi ) (13) As mentioned before, for mostly unknown reasons (see later), we came to the conclusion that the diﬀerence between my pure probabilistic integral and the real prime count seems to be constant and comes to a factor 1.7810292 if we want to correct for primes upto 1 billion. To illustrate the diﬀerence between my pure probabilistic integral and the logarithmic integral, we came to a factor 1.7810785 or 1/0.5614575. We will use the ﬁrst constant to optimize our prime count formula and the second to demonstrate our formula for the natural logarithm. The results are demonstrated in the table 1. 4.2 New formula for natural logarithm We were also eager to check our new formula for the natural logarithm and therefore we have added 2 more columns to our separate tables. One in which we approximate the logarithm based on our formula: we dividend our β with the total probability of all the primes until that prime. Another column in which we calculated the natural logarithm the standard way. A summary of this can be seen in table 2. 8
- 9. 4.2 New formula for natural logarithm 4 METHOD AND TECHNIQUES Table 1: In this table, we calculate the probability for a new prime immediately after another one (”My Integr”) and compare it with the logarithmic integral (”li(x)”) and with the perfect prime count (”Count”). My integr has been corrected with a factor α = 50847534/28549683 versus My pure. We can also deduct β = 28549683/50849229. Count P rime T ot prob M y pure M y integr li(x) 1 2 0.5 1 1 1 10 29 0.1579472231019522 6.82 11.37 11.77 100 541 0.0887498837753298 59 105 106 1000 7919 0.06246659294666621 569 1012 1015 10000 104729 0.04855897117165969 5632 10030 10037 100000 1299709 0.03988065693618553 56197 100088 100107 1000000 15485863 0.03391365652866189 561668 1000346 1000408 10000000 179424673 0.02954214659629333 5614822 10000162 10000496 20000000 373587883 0.02844462187012779 11230033 20001015 20001621 30000000 573259391 0.02784067067191051 16844593 30000712 30001582 40000000 776531401 0.02742791846692972 22459541 40001097 40002229 50000000 982451653 0.02711633109429194 28074042 50000689 50002077 50847534 999999937 0.02709315486988111 28549683 50847819 50849229 Table 2: In this table we calculate the natural logarithm of the respective prime using our new formula (”My ln(x)”) and compare it with the values using the standard formula (”ln(x)”). Count P rime M y ln(x) ln(x) 1 2 1.122915132600837 0.6931471805599453 10 29 3.554716286072388 3.367295829986474 100 541 6.326290721932071 6.293419278846481 1000 7919 8.988125329322656 8.977020214210413 10000 104729 11.56238595574076 11.55913134035915 100000 1299709 14.078443271354 14.07765095122063 1000000 15485863 16.55550075604224 16.55543810118943 10000000 179424673 19.00530702704133 19.00526602879025 20000000 373587883 19.73861944320852 19.73866383070949 30000000 573259391 20.16681181703334 20.16684886160037 40000000 776531401 20.47029441834519 20.47034763888541 50000000 982451653 20.70551374918884 20.70556169235444 50847534 999999937 20.72322581098074 20.72326577394641 9
- 10. 6 5 ACKNOWLEDGMENTS Discussions The question whether the lower part of the integral in formula 6 needs to start at 2 or maybe at 1 or 0 as well as the fact that the ﬁrst value ”1” in table 1 for ”MyIntegr” is the correct one can be answered as follows: In the range between 1 and before 2, the chance of ﬁnding a prime according to our logic is 100 percent. And our challenge only started at 1. So, the value for ”MyIntegr” could then be represented by the formula: π(x) = (x − 1). There is no need to have a correction factor here since this formula suits us ﬁne as the starting point π(1) = 0 and end point π(2) = 1 are correct according to our logic and according to the facts. The question of why we had to correct our probabilities with a factor α from 2 onward remains open. And to be honest, we can’t give a good mathematical answer. By using probabilities, we assumed that the probability of ﬁnding a prime is ﬂat over some range ∆(x). However, this ﬂat probability might not be the case over some longer distance. Immediately after a prime, we know that the next number will not be a prime since it will be even. However, there is a sort of higher chance that the number after this even number might be a prime. We could explain this by assuming that a bigger part of our periodic waves are still in the phase of going up or turning to come down but just not yet in the phase of touching the x-axis again. So we are actually always moving before the bigger waterfall of prime multiples that come down on us. It is logic from our side to try to optimize α according to our needs. In the ﬁrst part, we wanted to optimize it for getting a better prime counting formula, in the second part, we wanted to optimize it to come as close as possible to the natural logarithm. 6 Acknowledgments It was only after the writing of this article that we found out that earlier work of Harald Cram´r had similarities with the content of this article. However, what the Cram´r model e e didn’t have was our ”inﬁnitesimal” intake of using diﬀerent probabilities for each distance ∆(x) = pi − pi−1 as we are doing in formula 8 and 6. Instead, it seems, Cram´r used the same e √ probability for all p ≤ x. I would like to thank this publisher, his professional staﬀ and his volunteers for all the eﬀort they take in reading all the papers coming to them and especially I would like to thank this reader for reading my paper till the end. I would like to thank my wife for supporting me during the countless hours I spend behind my desk. 10
- 11. 7 7 REFERENCES References 1. Richard Crandall and Carl Pomerance (2005). Prime Numbers. Springer, second edition. Page 1,10. 11

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