Defining Prime Probability Analytically

Transcription

1 Defining Prime Probability Analytically An elementary probability-based approach to estimating the prime and twin-prime counting functions π(n) and π 2 (n) analytically Bhupinder Singh Anand Update of February 24, Abstract We define the residues r i (n) for all n 2 and all i 2 such that r i (n) = 0 if, and only if, i is a divisor of n; and show: (i) that M i = {(0, 1, 2,..., i 1), r i (n), 1 i } is a probability model for r i(n); and (ii) that the joint analytical probability P(r pi (n) = 0 r pj (n) = 0) of two primes p j dividing any integer n is the product P(r pi (n) = 0).P(r pj (n) = 0). We conclude that the analytical probability of n being a prime s given by the analytical prime probability distribution function P(n {p}) = π( n) i=1 (1 1 ) 2e γ log. en Hence the number π(n) of primes less than or equal to n is analytically approimated by π L (n) = n π( j) j=1 i=1 (1 1 ). We then show that, in the interval (p 2, n p2 ), the analytical approimation π n+1 L () of π() is a straight line with gradient n ); and that the function π L ()/ log is differentiable with derivative (π ()/ e L log e ) o(1). We conclude by the Law of Large Numbers that π() π L () since p 2 n+1 p2 ; and that both π ()/ n L log e and π()/ log e do not oscillate as. Chebyshev s Theorem, π() log e, then yields an elementary probability-based proof of the Prime Number Theorem π() log. We also show that the number π e 2 (n) of twin primes n is approimated by the analytical twin-prime counting function π T (n) = n j=1 P(j {p} j + 2 {p}); and conclude by the Law of Large Numbers that there are infinitely many twin primes since we show that π 2 (n) π T (n) e 2γ n. log 2 e n.1 1 Keywords: analytical prime counting function; analytical prime probability distribution function; 1

2 1 Introduction 1.1 Defining prime probability analytically Conventional wisdom appears to be that the distribution of primes suggested by the Prime Number Theorem, π(n) n log en, is such that the probability P(n {p}) of an integer n being a member of the set {p} of primes can only 1 be heuristically estimated as log en ; apparently reflecting an implicit faith in G. H. Hardy and J. E. Littlewood s 1922 dictum that 2 : Probability is not a notion of pure mathematics, but of philosophy or physics. It is a dictum which could reasonably be taken by the laity to also suggest, with some authority, that both a probability model for divisibility 3, and the probability P(n {p}) of an integer n being a prime p, are also not capable of being well-defined analytically 4 independently of the Theorem. Moreover whilst reasonably conceding 5 that the heuristic probability of an integer n being prime could also be naïvely assumed as n ) conventional wisdom seems to argue 6 against such naïvety by concluding that the number π(n) of primes less than or equal to n suggested by such probability would then be approimated by the heuristic function: π H (n) = n π( n) j=1 i=1 (1 1 ) = n. π( n) i=1 (1 1 ) 2.e γ n. log en For instance, Hardy and Littlewood note that: Chebyshev s Theorem; complete system of incongruent residues; de la Vallé Poussin; Dirichlect; Erdös; Euler s constant γ; Gauss; Hadamard; integer factorising algorithm; joint probability; Law of Large Numbers; Legendre; logarithmic integral function Li(); Mertens Theorem; mutually independent prime divisors; prime counting function π(n); Prime Number Theorem; probability model; probabilistic number theory; Selberg; twin-primes. MSC2010 Nos: 11A07, 11A41, 11A51, 11N36, 11Y05, 11Y11, 11Y16 2 [Gr95], p.19, fn.16 and p.20; see also [HL23], fn.4 on p.37, for the origin of the quote (courtesy Prof. Andrew Granville). 3 A probability model is a mathematical representation of a random phenomenon. It is defined by its sample space, events within the sample space, and a probability distribution function that defines the probabilities associated with each event in the sample space (see 8, Appendi III; also [El79a], Chapter 3, pp ). 4 See, for instance, [St02], Chapter 2, p.9, Theorem (sic) 2.1 (although [El79a] and [El79b] remain significantly silent on the issue). We use the term analytic for (nonheuristic) logical reasoning from first principles, accepted as being derivable transparently from the aioms and rules of inference of a mathematical language. 5 [Gr95], p Somewhat anomalously, as we argue in the Appendi in 7. 2

3 In the first place we observe that any formula in the theory of primes, deduced from considerations of probability, is likely to be erroneous in just this way. Consider, for eample, the problem what is the chance that a large number n should be prime? We know that the answer is that the chance is 1 approimately 7 log n. Now the chance that n should not be divisible by any prime less than a fied is asymptotically equivalent to (1 1 ϖ ) ϖ< and it would be natural to infer 1 that the chance required is asymptotically equivalent to But (1 1 ϖ ) ϖ< ϖ< (1 1 ϖ ) 2e C log n and our inference is incorrect, to the etent of a factor 2e C. 1 One might well replace ϖ < by ϖ <, in which case we should obtain a probability half as large. This remark is in itself enough to show the unsatisfactory character of the argument.... pp.36-37, G.H Hardy and J.E. Littlewood, Some problems of partitio numerorum: III: On the epression of a number as a sum of primes, Acta Mathematica, December 1923, Volume 44, pp However, the footnoted character of the argument can be considered unsatisfactory only if we conflate necessity with sufficiency! Otherwise, what we ought to reasonably conclude from the argument is that the probability that n should not be divisible by any prime ϖ less than is at most ϖ< (1 1 ϖ ), and necessarily eceeds ϖ< (1 1 ϖ )8. We shall show in this investigation that whilst the former product does, the latter product does not, define the analytic probabilities of the necessary 7 The word know suggests that this, curiously, can be viewed as an attempt to deduce beyond heuristics the approimate behaviour of the prime probability function P(n {p}) at finite n from the limiting behaviour of the prime counting function π(n) as n! 8 Or, for that matter, eceeds ϖ<n (1 1 ϖ ) for any n > (as noted in 3)! 3

4 and sufficient mutually independent conditions that define the distribution function 1 for r i i(n) over (0, 1, 2,..., i 1) which defines the primality of n under the probability model M i = {(0, 1, 2,..., i 1), r i (n), 1 } for divisibilty i defined in 2.1, where n + r i (n) 0 (mod i) and i > r i (n) 0. We shall further show the significance of such mutual independence for analytically estimating various prime counting functions; and consider some consequences for the prime and twin-prime counting functions π(n) and π 2 (n). 1.2 The functions π() and log e : A perspective To place this investigation in an appropriate historical perspective we note that Adrien-Marie Legendre and Carl Friedrich Gauss are reported 9 to have independently conjectured in 1796 that, if π() denotes the number of primes less than, then π() is asymptotically equivalent to. log e Between 1848/1850, Pafnuty Lvovich Chebyshev proved that π() and confirmed that if π()/ has a limit, then it must be log 110 e., log e The question of whether π()/ log e has a limit at all was answered in the affirmative first by Jacques Hadamard and Charles Jean de la Vallée Poussin independently in 1896, using argumentation involving functions of a comple variable 11 ; and again independently by Paul Erdös and Atle Selberg 12 in 1949/1950, using only elementary methods without involving functions of a comple variable. 1.3 The integral Li(): A perspective We also note that, reportedly 13 : In a handwritten note on a reprint of his 1838 paper Sur l usage des séries infinies dans la théorie des nombres, which he mailed to Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlect conjectured (under a slightly different form appealing to a series rather 9 cf. Prime Number Theorem. (2014, June 10). In Wikipedia, The Free Encyclopedia. Retrieved 09:53, July 9, 2014, from Prime number theorem&oldid= ; see also [Gr95]. 10 [Di52], p.439; see also [HW60], p.9, Theorem 7 and p.345, 22.4 for a proof of Chebychef s Theorem. 11 [Di52], p.439; see also [Ti51], Chapter III, p.8 for details of Hadamard s and de la Vallée Poussin s proofs of the Prime Number Theorem. 12 See [HW60], p.360, Theorem 433 for a proof of Selberg s Theorem. 13 cf. Prime Number Theorem. (2014, June 10). In Wikipedia, The Free Encyclopedia. Retrieved 09:53, July 9, 2014, from Prime number theorem&oldid=

5 than an integral) that an even better approimation to π() is given by the offset logarithmic integral Li() defined by: Li() = 2 1 log et.dt = li() li(2). We further note that in 1889 Jean de la Vallée Poussin proved 14 (cf. Fig.1):... that Li() represents π() more eactly than remaining approimations + log e log e (m 1)!. log 2 e loge m and its Fig.1: The asymptotic behaviour of the primes Fig.1: Graph showing ratio of the prime-counting function π() to two of its approimations, and Li(). As increases (note ais is logarithmic), both ratios tend towards ln 1. The ratio for converges from above very slowly, while the ratio for Li() converges ln more quickly from below The analytical probability P(n {p}) of n being a prime In this investigation we shall first define in 2 the residues r i (n) for all n 2 and all i 2 such that r i (n) = 0 if, and only if, i is a divisor of n 16. We shall then define the probability model M i = {(0, 1, 2,..., i 1), r i (n), 1} i for the residues r i (n) in 2.1, and show that the analytical probability P(p n) of the prime p dividing n is 1 ; and that the joint analytical probability p 14 [Di52], p cf. Prime Number Theorem. (2014, June 10). In Wikipedia, The Free Encyclopedia. Retrieved 09:53, July 9, 2014, from Prime number theorem&oldid= The residues r i (n) can also be graphically displayed variously as shown in the Appendi in 6. 5

6 P( n p j n) of two unequal primes, p j dividing any integer n is the product P( n).p(p j n) (in 2.2). We shall conclude in 3 that the analytical probability of n being a prime p is given by the analytical prime probability function 17 : P(n {p}) = π( n) i=1 (1 1 ) 2e γ log en. Fig.2: The graph of y = π( ) i=1 (1 1 ) Fig.2: Graph of y = P( {p}) = π( ) ). The dotted rectangles represent (p 2 j+1 p2 j ) j ) for j 1; and cumulatively add up to π L (). Figures within boes are values of the corresponding function within the interval (p 2, j p2 ) for j 2. j The analytical prime counting functions π L (n), π L () We shall then show in 3.2 that the number π(n) of primes p n is analytically approimated by the analytical prime counting function: 17 We show in 3.1 and in the Appendi 7 why we cannot further conclude from this that n. π( n) i=1 (1 1 ) estimates π(n) even heuristically. 6

7 π L (n) = n π( j) j=1 i=1 (1 1 ). We note in Fig.2 that the dotted areas representing (p 2 j+1 p2 j ) j ) for j 1 cumulatively yield π L () where, if p 2 n < p 2 n+1, we define the function π L () by: π L () = () = ( p 2 n ) n ) + n 1 j=1 (p2 j+1 p2 j ) j ) A probability-based proof that π()/ a limit log e tends to We shall then show in 4 that the graph (Fig.3) of π L () in the interval (p 2 n, p2 n+1 ) for n 1 is a straight line with gradient n ). Fig.3: The graph of y = () = π L () Fig.3: y = π L () = ( p 2 n ) n i=1 (1 1 )+ n 1 j=1 (p2 j+1 p2 j ) j i=1 (1 1 )+2 in the interval (p 2 n, p2 n+1 ). We shall further show in 4.1 that the function π L ()/ log e within the interval (p 2, n p2 ), and that (the derivative): n+1 is differentiable 7

8 (π L ()/ log e ) o(1) We shall thus conclude that the function π L ()/. log e does not oscillate as We shall further conclude in 4.2 by the Law of Large Numbers that π() π L () since p 2 p2 ; and that both π ()/ n+1 n L log e and π()/ do log e not oscillate as, but tend to a limit. Chebyshev s Theorem that π() log e then yields an elementary probability-based proof of the Prime Number Theorem π(). log e 1.7 The twin-prime counting function π T (n) We demonstrate the broader significance, of defining the probability of n being a prime analytically, by showing in 5 that the number π 2 (n) of twin primes n is analytically approimated by the analytical twin-prime counting function: π T (n) = n j=1 P(j {p} j + 2 {p}) and concluding, by the Law of Large Numbers, that there are infinitely many twin-primes since we show that π 2 (n) π T (n) e 2γ n. 2 The residues r i (n) log 2 e n. We begin by defining the residues r i (n) for all n 2 and all i 2 as below 18 : Definition 1 n + r i (n) 0 (mod i) where i > r i (n) 0. Since each residue r i (n) cycles over the i values (i 1, i 2,..., 0), these values are all incongruent and form a complete system of residues 19 mod i. It immediately follows that: Lemma 1 r i (n) = 0 if, and only if, i is a divisor of n. 2.1 The Probability Model M i = {(0, 1, 2,..., i 1), r i (n), 1 i } By the standard definition of the distribution function P(e) for the probability of an event e 20, we conclude for any i 2 that M i = {(0, 1, 2,..., i 18 The residues r i (n) can also be graphically displayed variously as shown in the Appendi in [HW60], p See 8, Appendi III; also [Ko56], Chapter I, 1, Aiom III, p.2; [El79b], Chapter 13, pp

9 1), r i (n), 1 i } is a probability model 21 for the values of r i (n) over the sample space (0, 1, 2,..., i 1) for n 2, since we have by Lemma 1 that: Lemma 2 For any n 2, i 2 and any given integer i > u 0, the probability P(r i (n) = u) that r i (n) = u is 1, u=i 1 i u=0 P(r i (n) = u) = 1, and the probability P(r i (n) u) that r i (n) u is 1 1. i Corollary 1 For any n 2 and any prime p 2, the probability P(r p (n) = 0) that r p (n) = 0, and that p divides n, is 1 ; and the p probability P(r p (n) 0) that r p (n) 0, and that p does not divide n, is 1 1. p 2.2 The prime divisors of any integer n are mutually independent We note the standard definition 22 : Definition 2 Two events e i and e j are mutually independent for i j if, and only if, P(e i e j ) = P(e i ).P(e j ). We then have that: Lemma 3 If and p j are two primes where i j then, for any n 2, we have: P((r pi (n) = u) (r pj (n) = v)) = P(r pi (n) = u).p(r pj (n) = v) where > u 0 and p j > v 0. Proof : The.p j numbers v. + u.p j, where > u 0 and p j > v 0, are all incongruent and form a complete system of residues 23 mod (.p j ). Hence: P((r pi (n) = u) (r pj (n) = v)) = 1.p j By Lemma 2: P(r pi (n) = u).p(r pj (n) = v) = ( 1 )( 1 p j ). 21 Defined formally in 8, Appendi III. 22 See 8, Appendi III; also [Ko56], Chapter VI, 1, Definition 1, p.57 and 2, p.58; [El79a], p.29; [Ka59], p [HW60], p.52, Theorem 59. 9

10 The lemma follows. If u = 0 and v = 0 in Lemma 3, so that both and p j are prime divisors of n, we immediately conclude by Definition 2 that: Corollary 2 P((r pi (n) = 0) (r pj (n) = 0)) = P(r pi (n) = 0).P(r pj (n) = 0). We can also epress this as: Corollary 3 P( n p j n) = P( n).p(p j n). We thus conclude that 24 : Theorem 1 The prime divisors of any integer n are mutually independent. 3 The analytical probability P(n {p}) that n is a prime Prima facie, conventional wisdom is apparently that the distribution of primes suggested by the Prime Number Theorem (PNT) is such that the probability P(n {p}) of n being a prime p may be heuristically assumed as ; log en 1 apparently reflecting an implicit belief that such probability is not capable of being well-defined analytically independently of the Theorem 25. However, since n is a prime if, and only if, it is not divisible by any prime p n, it follows immediately from Lemma 2 and Lemma 3 that: Lemma 4 For any n 2, the probability P(n {p}) of an integer n being a prime s the probability that r pi (n) 0 for any 1 i k if p 2 k n < p2 k+1. By Corollary 1 we can epress this by the analytical prime probability function (graphically illustrated in 1.6, Fig.2) 26 : Theorem 2 P(n {p}) = π( n) i=1 (1 1 ) 2e γ log en. 24 In a companion paper we show how it immediately follows from Theorem 1 that integer factorising is necessarily of order O(n/log e n); from which we conclude that integer factorising cannot be in the class P of polynomial-time algorithms. 25 See for instance [Gr95], p We note that Lt n log e n. π( n) i=1 (1 1 ) = 2.e λ ([Gr95], p.13). 10

11 3.1 The heuristic function π H (n) We also note prima facie that conventional wisdom whilst reasonably arguing 27 that the heuristic probability of a number n being prime may also be assumed as π( n) i=1 (1 1 ) appears to anomalously argue that π(n) as suggested by such probability would be heuristically approimated by the heuristic function 28 : π H (n) = n j=1 π( n) i=1 (1 1 ) = n. π( n) i=1 (1 1 ) 2.e γ n log en. 3.2 The analytical prime counting function π L (n) However, it follows from Theorem 2 that, since p 2 p2 as n, n+1 n by the Law of Large Numbers 29, an analytical estimate 30 of the number π(n) of primes less than or equal to n is the analytical prime counting function (graphically illustrated in 1.4, Fig.3): Definition 3 π L (n) = n π( j) j=1 i=1 (1 1 ). 4 The interval (p 2 n, p2 n+1 ) We note that it follows immediately from the definition of π() as the number of primes less than or equal to that: Lemma 5 π( ) i=1 (1 1 ) = π( +1) i=1 (1 1 ) for p 2 n < p 2 n+1. We can also generalise the number-theoretic function of Definition 3 as: Definition 4 π L () = π L (p 2) + ( n p2) n n ) for p 2 n < p 2 n+1. We note that the graph of π L () in the interval (p 2, n p2 ) for n 1 is now n+1 a straight line with gradient n ), as illustrated in 1.6, Fig.3 where we defined π L () equivalently by: π L () = () = ( p 2 n ) n ) + n 1 j=1 (p2 j+1 p2 j ) j ) See for instance [Gr95], p Anomalously since (as may be seen in Fig.2), if π H (n) were a valid approimation to π(n), then the probability of k being a prime for any given k would tend to 0 as n! Specifically (as we note in the Appendi in 7) if we treat π H (n) as a prime counting function, then an apparent anomaly surfaces when we epress π H (n) and π(n) in terms of the number of primes determined by each function respectively in each interval (p 2 n, p2 n+1 ). 29 See 8, Appendi III; also [Ko56], Chapter VI, 3, p.61; [El79b], pp i.e. epected value : see 8, Apendi III. Compare also [HL23], pp

12 4.1 The function π L ()/ log e We consider net the function π L ()/ log e in the interval (p2 n, p2 n+1 ): π L ()/ log e = (π L (p2 n ) + ( p2 n ) n ))/ log e This now yields the derivative (π L (). loge ) in the interval (p 2, n p2 ) as: n+1 π L ().( loge ) + (π L ()). loge (π L (p 2 n )+( p2) n n i=1 (1 1 )).( loge ) +(π L (p 2 n )+( p2) n n i=1 (1 1 )). loge (π L (p 2) + ( n p2) n n )).( 1 loge ) + ( n 2 2 )). loge Since p 2 n < p 2 n+1 and π L () π() by the Law of Large Numbers, by Mertens 31 and Chebyshev s Theorems we can epress the above as: (π L (p 2 n ) + e γ ( p 2 n ) log en ( π L (p2 n ) ).( 1 loge ) + e γ.log e 2 2.log en + e γ (1 p2 n log en )). (1 loge) + e γ.log e.log en ( π L (p2 n ). p2 p 2 n n + e γ (1 p2 n log en )). (1 2.logep n) + 2.e γ.log ep n p 2 p n 2 n.logen Since each term 0 as n, we conclude that the function π L ()/ log e does not oscillate but tends to a limit as since: Lemma 6 (π L ()/ log e ) o(1). 4.2 A probability-based proof of the Prime Number Theorem The above now yields a probability-based proof that: Theorem 3 π()/ log e tends to a limit. Proof : By Lemma 6 the function π L ()/ but tends to a limit as. log e does not oscillate Since p 2 p2 as n, and π() π n+1 n L() by the Law of Large Numbers, the theorem follows from Chebyshev s Theorem that π() /log e. 31 [HW60], Theorem 429, p

13 5 The significance of defining prime probability analytically: A probability-based proof that there are infinitely many twin-primes We net demonstrate that, by Theorem 2, we can define the twin-prime counting function π T (n), which analytically estimates the number π 2 (n) of twin primes (, +1 = + 2) for 3 n as: Definition 5 π T (n) = n j=1 P(j {p} j + 2 {p}) In order to estimate π T (n), we first define: Definition 6 An integer n is a TW integer if, and only if, r pi (n) 0 and r pi (n) 2 for all 1 i π( n). Since n is a prime if, and only if, it is not divisible by any prime p n, we then have that: Lemma 7 If n is a TW integer, then n is a prime. Proof : The lemma follows immediately from Definition 6, Definition 1 and Lemma 1. Lemma 8 If n is a TW integer, then n + 2 is either a prime or p 2 π( n)+1. Proof : By Definition 6 and Definition 1: r pi (n) 2 for all 1 i π( n) n + 2 λ.i for all 2 i p π( n), λ 1 Hence, if n + 2 is divisible by p π( n)+1, then n + 2 = p 2 ; else π( n)+1 it is a prime. Since each residue r i (n) cycles over the i values (i 1, i 2,..., 0), these values are all incongruent and form a complete system of residues mod i. It thus follows from Definition 6 and Section 2.1 that the probability of n 9 being a TW integer is: Lemma 9 P(n {TW}) = π( n) i=2 (1 2 ) 13

14 The number π TW (n) of TW integers 9 but n is thus: Lemma 10 π TW (n) = n π( j) j=9 i=2 (1 2 ) Since the number of TW integers such that n + 2 = p 2 π( n)+1 π( n), it also follows that, for n 9: Lemma 11 π T (n) n π( j) j=9 i=2 (1 2 ) π( n). We further note that: Theorem 4 π T (n) as n. Proof 32 : We have by Lemma 11 that, for n 9: π T (n) (n 9). π( n) i=2 (1 2 ) π( n) (n 9). π( n) i=2 (1 1 )(1 1 ( 1) ) π( n) (n 9). π( n) i=2 (1 1 )(1 1 1 ) π( n) (n 9). π( n) i=2 (1 1 1 ) 2 π( n) is not more than (n 9). n ) 2 π( n) Now, by Chebyshev s and Mertens Theorems, we have that: (n 9). n ) 2 π( n) (n 9).( e γ log en )2 π( n) The theorem follows. e 2γ. n log 2 e n 9e 2γ log 2 e n O( n log en ) as n Since p 2 p2 as n, it follows by the Law of Large Numbers n+1 n that π 2 (n) π T (n) π T W (n). We conclude that there are infinitely many twin primes, and that 33 : Corollary 4 π 2 (n) e 2γ. n. loge 2 n 32 We note that the following argument is a special case of the limiting behaviour of the Generalised Prime Counting Function in 9, Theorem Where e 2γ = ; compare [HW60], p.371, 22.20: π 2 (n) 2C 2. n where C 2 = p 3 p(p 2) (p 1) log 2 e n,

15 6 Appendi I: The residue function r i (n) In this Appendi we graphically illustrate how the residues r i (n) occur naturally as values of: A: The natural-number based residue functions R i (n); B: The natural-number based residue sequences E(n); and as the output of: C: The natural-number based algorithm E N ; D: The prime-number based algorithm E P ; E: The prime-number based algorithm E Q. A: The natural-number based residue functions R i (n) The residues r i (n) can be defined for all n 1 as the values of the naturalnumber based residue functions R i (n), defined for all i 1 as below in Fig.4. We note that each function R i (n) cycles through the values (i 1, i 2,..., 0) with period i. Fig.4: The natural-number based residue functions R i (n) Function: R 1 n R 2 n R 3 n R 4 n R 5 n R 6 n R 7 n R 8 n R 9 n R 10 n R 11 n... R n n n = n-1 n = n-2 n = n-3 n = n-4 n = n-5 n = n-6 n = n-7 n = n-8 n = n-9 n = n-10 n = n-11 n r 1 n r 2 n r 3 n r 4 n r 5 n r 6 n r 7 n r 8 n r 9 n r 10 n r 11 n... 0 Fig.4: The natural-number based residue functions R i (n) 15

16 B: The natural-number based residue sequences E(n) The above residues r i (n) can also be viewed alternatively as values of the associated residue sequences, E(n) = {r i (n) : i 1}, defined for all n 1, as illustrated below in Fig.5. We note that: The sequences highlighted in red identify a prime 34 p (since r i (p) 0 for 1 < i < p); The boundary residues r 1 (n) = 0 and r n (n) = 0 are identified in blue. Fig.5: The natural-number based residue sequences E(n) Function: R 1 n R 2 n R 3 n R 4 n R 5 n R 6 n R 7 n R 8 n R 9 n R 10 n R 11 n... R n n E(1): n-1 E(2): n-2 E(3): n-3 E(4): n-4 E(5): n-5 E(6): n-6 E(7): n-7 E(8): n-8 E(9): n-9 E(10): n-10 E(11): n E(n): r 1 n r 2 n r 3 n r 4 n r 5 n r 6 n r 7 n r 8 n r 9 n r 10 n r 11 n Fig.5: The natural-number based residue sequences E(n) C: The output of a natural-number based algorithm E N We give below in Fig.6 the output for 1 n 11 of a natural-number based algorithm E N that computes the values r i (n) of the sequence E N (n) for only 1 i n for any given n. 34 Conventionally defined as integers that are not divisible by any smaller integer other than 1. 16

17 Fig.6: The output of the natural-number based algorithm E N Divisors: n... E N (1): 0 E N (2): 0 0 E N (3): E N (4): E N (5): E N (6): E N (7): E N (8): E N (9): E N (10): E N (11): E N (n): r 1 n r 2 n r 3 n r 4 n r 5 n r 6 n r 7 n r 8 n r 9 n r 10 n r 11 n Fig.6: The output of the natural-number based algorithm E N D: The output of the prime-number based algorithm E P We give below in Fig.7 the output for 2 n 31 of a prime-number based algorithm E Q that computes the values q i (n) = r pi (n) of the sequence E P (n) for only each prime 2 n for any given n. Fig. 7: The output of the prime-number based algorithm E P Prime: p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p p n... Divisor: p n... E P (2): 0 E P (3): 1 0 E P (4): 0 2 E P (5): E P (6): E P (7):

18 E P (8): E P (9): E P (10): E P (11): E P (12): E P (13): E P (14): E P (15): E P (16): E P (17): E P (18): E P (19): E P (20): E P (21): E P (22): E P (23): E P (24): E P (25): E P (26): E P (27): E P (28): E P (29): E P (30): E P (31): E P (n): q 1 n q 2 n q 3 n q 4 n q 5 n q 6 n q 7 n q 8 n q 9 n q 10 n q 11 n Fig.7: The output of the prime-number based algorithm E P E: The output of the prime-number based algorithms E P and E Q We give below in Fig.8 the output for 2 n 121 of the two prime-number based algorithms E P (whose output {q i (n) = r pi (n) : 1 i π(n)} is shown only partially, partly in gray) and E Q (whose output q i (n) = {r pi (n) : 1 i π( n)} is highlighted in black and red, the latter indicating the generation of a prime sequence and, ipso facto, definition of the corresponding prime For informal reference and perspective, formal definitions of both the prime-number based algorithms E P and E Q are given in this work in progress Factorising all m n is of order Θ( n i=2 π( i)). 18

19 Fig.8: The output of the prime-number based algorithms E P and E Q Prime: p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p p n... Divisor: p n... Function: Q 1 n Q 2 n Q 3 n Q 4 n Q 5 n Q 6 n Q 7 n Q 8 n Q 9 n Q 10 n Q 11 n... E Q (2): 0 (Prime by definition) E Q (3): 1 0 E Q (4): 0 2 E Q (5): E Q (6): E Q (7): E Q (8): E Q (9): E Q (10): E Q (11): E Q (12): E Q (13): E Q (14): E Q (15): E Q (16): E Q (17): E Q (18): E Q (19): E Q (20): E Q (21): E Q (22): E Q (23): E Q (24): E Q (25): E Q (26): E Q (27): E Q (28): E Q (29): E Q (30): E Q (31): E Q (32): E Q (33): E Q (34):

20 E Q (35): E Q (36): E Q (37): E Q (38): E Q (39): E Q (40): E Q (41): E Q (42): E Q (43): E Q (44): E Q (45): E Q (46): E Q (47): E Q (48): E Q (49): E Q (50): E Q (51): E Q (52): E Q (53): E Q (54): E Q (55): E Q (56): E Q (57): E Q (58): E Q (59): E Q (60): E Q (61): E Q (62): E Q (63): E Q (64): E Q (65): E Q (66): E Q (67): E Q (68): E Q (69): E Q (70): E Q (71): E Q (72): E Q (73): E Q (74): E Q (75):

21 E Q (76): E Q (77): E Q (78): E Q (79): E Q (80): E Q (81): E Q (82): E Q (83): E Q (84): E Q (85): E Q (86): E Q (87): E Q (88): E Q (89): E Q (90): E Q (91): E Q (92): E Q (93): E Q (94): E Q (95): E Q (96): E Q (97): E Q (98): E Q (99): E Q (100): E Q (101): E Q (102): E Q (103): E Q (104): E Q (105): E Q (106): E Q (107): E Q (108): E Q (109): E Q (110): E Q (111): E Q (112): E Q (113): E Q (114): E Q (115): E Q (116):

22 E Q (117): E Q (118): E Q (119): E Q (120): E Q (121): E Q (n): q 1 n q 2 n q 3 n q 4 n q 5 n q 6 n q 7 n q 8 n q 9 n q 10 n q 11 n Prime: p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p p n... Divisor: p n... Fig.8: The output of the prime-number based algorithms E P and E Q 7 Appendi II: Why π H (n) is not a prime counting function We note an apparent anomaly that surfaces when we epress π(n), π L (n), and the function π H (n) defined in 3.1, in terms of the number of primes determined by each function respectively in each interval (p 2, n p2 ) as follows: n+1 π(p 2 n+1 ) = n j=1 (π(p2 j+1 ) π(p2 j )) + π(p2 1 ) π L (p 2 n+1 ) = n j=1 (π L (p2 j+1 ) π L (p2 j )) + π L (p2 1 ) π H (p 2 n+1 ) = p2 n+1. π( p 2 n+1 ) ) = ( n j=1 (p2 j+1 p2 j ) + p2 1 ). n ) = n j=1 (p2 j+1. n ) p 2 j. n )) + p 2 1. n ) Reason: By 3.2 π L (n) is an analytical estimate of π(n), and the n th term: π L (p 2 ) π k+1 L (p2 ) > 0 for any given k k whilst, as n, the n th term: p 2 k+1. n ) p 2 k. n ) 0 for any given k Compare with [St02], Chapter 2, p.9, Theorem (sic) 2.1, where the author essentially concludes from a similar Proof via contradiction that any function such as π L (n) cannot be an analytical estimate of π(n), and so there can be no Probability Model for Divisibility! 22

23 More specifically, by 3.2 and Mertens theorem, the analytical estimate of the number of primes between the prime squares p 2 k+1 and p2 k (see fig.2), for any k > 1, is given by 37 : π L (p 2 ) π k+1 L (p2) = (p2 p2). k k k+1 k ) ((p k + 2) 2 p 2). k k ) 4(p k + 1). k ) 4p k (1 + 1 ). k p k ) O( p k log ep k ) Moreover, if we were to accept π H (n) also as an analytical prime counting function, then an apparent anomaly follows from the Prime Number Theorem π(n) n since π (n) 2.e γ n! log en H log en 8 Appendi III: Definitions of some terms and concepts of Probability Theory Probability model 38 : A probability model is a mathematical representation of a random phenomenon. It is defined by its sample space, events within the sample space, and a distribution function that defines the probabilities associated with each event. The sample space S for a probability model is the set of all possible outcomes. An event A is a subset of the sample space S. A probability is a numerical value assigned to a given event A. Distribution Function 39 : Let X be a random variable which denotes the value of the outcome of a certain eperiment, and assume that this eperiment has only finitely many possible outcomes. Let Ω be the sample space of the eperiment (i.e., the set of all possible values of X, or equivalently, the set of all possible outcomes of the eperiment). A distribution function for X is a real-valued function m whose domain is Ω and which satisfies: 37 Compare Brocard s conjecture: π(p 2 k+1 ) π(p2 k ) cf. Finite Probability Spaces in [El79a], Chapter 3, pp Ecerpted from [GS97], Chapter 1, 1.2, p

24 1. m(ω) 0, for all ω n, and 2. ω Ω m(ω) = 1. For any subset E of Ω, we define the probability of E to be the number P (E) given by P (E) = ω E m(ω) Some notations 40 : Let A and B be two sets. Then the union of A and B is the set A B = { A or B} The intersection of A and B is the set The difference of A and B is the set A B = { A and B} A B = { A and / B} The set A is a subset of B, written A B, if every element of A is also an element of B. Finally, the complement of A is the set A = { Ω and / A}. Mutual Independence 41 : A set of events {A 1, A 2,..., A n } is said to be mutually independent if for any subset {A i, A j,..., A m } of these events we have P (A i A j... A m ) = P (A i )P (A j )... P (A m ), or equivalently, if for any sequence A 1, A 2,..., A n with A j = A j or A j, P (A i A j... A m ) = P (A i )P (A j )... P (A m ). Epected Value 42 : Let X be a numerically-valued discrete random variable with sample space Ω and distribution function m(). The epected value E(X) is defined by: 40 Ecerpted from [GS97], Chapter 1, 1.2, p Ecerpted from [GS97], Chapter 4, 4.1, Definition 4.2, p Ecerpted from [GS97], Chapter 5, 5.1, p

25 E(X) = m(), Ω provided this sum converges absolutely. We often refer to the epected value as the mean, and denote E(X) by µ for short. If the above sum does not converge absolutely, then we say that X does not have an epected value. Law of Large Numbers 43 : Let X 1, X 2,..., X n be an independent trials process, with finite epected value µ = E(X j ) and finite variance σ 2 = V (X j ). Let S n = X 1 + X X n. Then for any ɛ > 0, P ( Sn n µ ɛ) 0 as n. Equivalently, P ( Sn n µ < ɛ) 1 as n. 9 Appendi IV: The Generalised Prime Counting Function: n π( j) j=1 i=a (1 b p ) i We note that the argument of Theorem 4 in 5 is a special case of the limiting behaviour of the Generalised Prime Counting Function n π( j) j=1 i=a (1 b ), which estimates the number of integers n such that there are b values that cannot occur amongst the residues r pi (n) for a i π( j) 44 : Theorem 5 n j=1 Proof : For p a > b 1, we have that: π( j) i=a (1 b ) as n if p p a > b 1. i n π( j) j=1 i=a (1 b ) n j=p 2 a π( j) i=a (1 b ) n j=p 2 a π( n) i=a (1 b ) (n p 2). π( n) a i=a (1 b ) (n p 2 a ). n i=a (1 b ) 43 Ecerpted from [GS97], Chapter 8, 8.1, p.307, Theorem Thus b = 1 yields an estimate for the number of primes n, and b = 2 an estimate for the number of TW primes (Definition 6) n. 25

26 The theorem follows if: log e (n p 2 a ) + n i=a log e(1 b ) (i) We note first the standard result for < 1 that: log e (1 ) = m m=1 m For any > b 1, we thus have: Hence: log e (1 b ) = (b/ ) m m=1 m = b (b/ ) m m=2 m n i=a log e(1 b ) = n i=a ( b ) n i=a ( (b/ ) m m=2 ) m (ii) We note net that, for all i a: c < (1 b p a ) c < (1 b ) It follows for any such c that: (b/ ) m m=2 m m=2 ( b ) m = (b/ )2 1 b/ b2 c.p 2 i Since: i=1 1 p 2 i = O(1) it further follows that: n i=a ( m=2 (b/ ) m (iii) From the standard result 45 : m ) n i=a ( b2 ) = O(1) c.p 2 i 1 p = log p elog e + O(1) + o(1) it then follows that: n i=a log e(1 b ) n i=a ( b ) O(1) b.(log e log e n + O(1) + o(1)) O(1) 45 [HW60], p.351, Theorem

27 The theorem follows since: log e (n p 2 a ) b.(log elog e n + O(1) + o(1)) O(1) and so: log e (n p 2 a ) + n i=a log e(1 b ) Acknowledgements I am indebted to my erstwhile classmate, Professor Chaitanya Kumar Harilan Mehta, for his unqualified encouragement and support for my scholarly pursuits over the years; most pertinently for his patiently critical insight into, and persistent insistence upon, the required rigour for defining the probability of a number being prime analytically, without which this etension of a 1964 investigation into the nature of divisibility and the structure of the primes begun whilst yet classmates would have vanished into some black hole of the informal universe of seemingly self-evident truths. References [Di52] [El79a] [El79b] [Gr95] [GS97] [HW60] [HL23] Leonard Eugene Dickson History of the Theory of Numbers: Volume I. Chelsea Publishing Company, New York, N. Y. P. D.T. A. Elliott Probabilistic Number Theory I. Springer-Verlag, New York. P. D.T. A. Elliott Probabilistic Number Theory II. Springer-Verlag, New York. Andrew Granville Harald Cramér and the distribution of prime numbers. Scandinavian Actuarial Journal, Volume 1995, Issue 1, pp DOI: / Charles M. Grinstead and J. Laurie Snell Introduction to Probability, The CHANCE Project Version dated 4 July 2006 of the Second Revised Edition, 1997, American Mathematical Society, Rhode Island, USA. G. H. Hardy and E. M. Wright An Introduction to the Theory of Numbers. 4th edition. Clarendon Press, Oford. G.H Hardy and J.E. Littlewood Some problems of partitio numerorum: III: On the epression of a number as a sum of primes, Acta Mathematica, December 1923, Volume 44, pp [Ka59] Mark Kac Statistical Independence in Probability, Analysis and Number Theory The Carus Mathematical Monographs: Number Twelve The Mathematical Association of America, Second Impression, [Ko56] [St02] A. N. Kolmogorov Foundations of the Theory of Probability. Second English Edition. Translation edited by Nathan Morrison Chelsea Publishing Company, New Yourk (sic). Jörn Steuding Probabilistic Number Theory. The Pennsylvania State University Cite- SeerX Archives, doi=

28 [Ti51] E. C. Titchmarsh The Theory of the Riemann Zeta-Function. Clarendon Press, Oford. Authors postal address: #1003 B Wing, Lady Ratan Tower, Dainik Shivner Marg, Worli, Mumbai , Maharashtra, India. bhup.anand@gmail.com. Mbl: