Quantum and classical study of prime numbers, prime gaps and their dynamics

A wave function constructed from prime counting function is employed to study the properties of primes using quantum dynamics. The prime gaps are calculated from the expectation values of position and a formula for maximal gaps is proposed. In an analogous nonlinear system, the trajectories, associated nodes with their stability condition and the bifurcation dynamics are studied using classical dynamics. It is interesting to note that the Lambert W functions appear as a natural solution for the fixed points as functions of energy. The derived potential with the divergence resembles the effective potential experienced by a particle near a massive spherical object in general theory of relativity. The coordinate time and proper time corresponding to a black hole serendipitously find their analogy in the solution of the nonlinear dynamics representing primes. The stereographic projection obtained from quantum dynamics on unit circle in the (θ,pθ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\theta ,p_{\theta })$$\end{document} phase space of the real numbers present along x-axis in general and prime numbers in particular provides a simple way to calculate a formula for upper bounds on the prime gaps. The estimated prime gaps is found to be significantly better than that of Cramer’s predicted values.


Introduction
"Can we formulate and study pure numbers particularly the prime numbers (which are integers that are divisible by 1 and by themselves) as a dynamical problem in physics ?" In search of an affirmative answer, a lot of efforts [1][2][3][4][5][6][7] have been made to link the prime number theory to some measurable parameters in various branches of physics like classical mechanics, quantum mechanics, statistical physics and spectroscopy etc. It is worth pointing here that, the nontrivial zeros of the number theoretic function, Riemann zeta function is intimately related to the prime numbers. The Riemann zeta function is the analytic continuation of the function ζ(k) constructed from the prime numbers p n through the relation [8] ζ(k) = p n 1 − 1 p k n −1 . Wolf [8] and Schumayer and Hutchinson [9] provide exhaustive reviews with critical analysis of the different approaches and models used by various authors on the linkage with the number theory [10]. In quantum mechanics, Hilbert-Polya conjecture [8] relates the nontrivial zeros of the Riemann zeta function with the real eigenvalues of a Hermitian operator,or more specifically the energy levels corresponding to a self adjoint Hamiltonian of a quantum system. Accordingly Hamiltonian for many one-dimensional quantum systems were proposed wherein the energy eigenvalues mimic the nontrivial zeros of the Riemann zeta function. Recently Bender, Brody and Muller constructed [2] a Hamiltonian operatorĤ which is the unitary transformation of the symmetrized Berry and Keating HamiltonianĤ = x p + px. He proposed that under certain boundary condition the eigen value of theĤ whose classical limit is 2x p corresponds to nontrivial zeros of the Riemann-zeta function. However some limitations on this theory has been pointed out in the review of M. Wolf [8]. Here we construct a different problem relating to primes by using a single particle (quantum) mechanical system. A local single particle probability density is calculated from the asymptotic form of the prime counting function. Using the square root of the single particle probability density as a wave function in the Schrodinger equation, we determine the potential. It is interesting to note that a formula for the prime gap which is the difference between two successive primes is obtained by considering the expectation value of x. Subsequently we extend the derivation to propose an approximate but robust formula for the maximal prime gap which shows that the primes become rarer as we move on the x-axis towards x → ∞. Using the interaction potential, we have analyzed the classical dynamics of the particles, which reveal indirectly the non-linear dynamical [11] aspects that one can associate with the prime numbers. We find that the prime number system depicts a blue-sky bifurcation and the solutions for the fixed points as functions of energy are related to the Lambert W functions [12]. This nonlinear dynamics of the classical problem provides an example where the Lambert W function appears naturally [13]. The fact that prime counting function can be cast in the form of Lambert W function which also appears [14,15] in the solution of a two body problem in general theory of relativity, provides a connection to black hole [7,16] physics which is of much transcendence. We relate our solution of fixed points in the classical nonlinear dynamics of the single particle under study to the coordinate and proper time associated with a black hole. The stereographic projection [17,18] of the real numbers on the x-axis in general and prime numbers in particular provides a simple way to calculate a formula for an upper limit on the prime gap as well. The classical and the quantum mechanical problems associated with the Schrodinger equation on the unit circle are also analysed. The organization of the paper is as follows: in Sec-II and Sec-III we study the system representing primes in x − p x space and in Sec-IV in θ and p θ space. In Section II, we have presented the quantum mechanical analysis and derived the potential. Specifically in sec-IIA the wave function is proposed from prime density and in sec-IIB the prime gap is formulated on the basis of the expectation value of x. In Sec-III, the classical aspects of the dynamical problem associated with the prime numbers are analysed on the basis of the interaction potential (A) Phase space, trajectories and zeroes (B) bifurcation phenomena and the distance between the two zeros and (C) relevance to the dynamics of Black Hole (D). We have added Appendix-A wherein the comparison of the potential corresponding to gravity and the prime potential and in Appendix-B the aspects of four acceleration [19] resolving the divergence in the potential. In Section-IV we treat the problem in θ − p θ space, (A) derive prime gap using Stereographic projection, (B) study the dynamics through the classical Hamiltonian in θ coordinate on the unit circle, (C) construct the quantum Hamiltonian and establish the relation between the expectation value and the prime gaps.

Quantum mechanical analysis in x − p x space
This section deals with the quantum dynamics of a single particle [20,21] moving on x-axis in the background potential V (x). To study such system, we write down the one particle Schrodinger equation as Usually in quantum mechanics the energy eigen values and corresponding wave functions are developed from the Schrodinger equation for a known potential [22]. In contrast by using the prime counting function which gives the number of primes below x, here we construct a probability density associated with the prime as a function of x in Fig. 1 The plot of density ρ(x) as a function of x. ρ(x) is negative for 0 < x < 1 ,diverges at x = 1 and positive for x > 1 one dimension and there from estimate the wave function. We aim to find a formula for the prime gaps and also to determine the effective potential to study the corresponding classical single particle in the phase space to predict certain properties of primes.

Construction of probability density and wave function for primes
In this section we construct the eigen function ψ(x) of Schrodinger Eq. (1) from π(x), where π(x) is known as prime counting function which gives the number of primes below the real number x. For example if x = 4, then π(x) = 2 that is 2 and 3 are the two prime numbers below 4. According to prime number theorem, the asymptotic distribution [23] of primes appears as x ln(x) , (π(x)/(x/ln(x)) → 1 as x → ∞). We denote π a (x) = x ln(x) and use this function in our calculation to find a characteristic single particle local density of prime numbers in one dimension as ρ(x) = π a (x) ln(x) . This gives a distribution of the prime numbers below each value of x, due to the fact that π(x) which is actually a stair case function, with unit steps at the primes is taken [1] at the roughest level of description as a smoothed one as π a (x) = x ln(x) by ignoring the variations in the prime counting function. Taking Born's interpretation [24] of probability density into account, the single particle wave function ψ(x) can be constructed from the density function as The above wave function though vanishes at infinity, that is ψ(x) → 0 as x → ∞, can be normalized by taking the limits between the lowest prime number 2 and ∞. Since we study a single particle system, the normalization of the wave function can be written as On the substitution of wave function as explained in Eq. (2), [22] the potential function can be determined (up to an additional constant and without loss of generality we have taken E = 0) as It is worth pointing here that Heisenberg uncertainty principle leads to infinitesimal energy entity for systems having infinite uncertainty in particle position and time. Equivalently, it can be said that infinite freedom of space and time is a characteristic feature of zero energy entity [25].
At x = 1 where prime numbers don't exist, both the potential and the wave function diverges. However for the domain of the prime numbers between 2 to ∞, the range of potential decreases from 0.2853 to 0 in the units of h, m = 1 and the wave function except the normalization constant decrease from 1.201 to 0. This potential will be latter used to study the hidden classical nonlinear dynamics of the system representing primes.

Prime gap and expectation value of x
In this section we calculate the expectation value of x over the wave function given in Eq. (2) and show that this is another way of calculating the prime gaps given as p n+1 − p n where p n is the nth prime number. Though one can study prime gaps using the prime counting function and without use of physical models, here we propose an interesting path to calculate prime gaps by taking quantum-mechanical formalism based on expectation value. The non-monotonic nature [26] of the prime gaps gets reflected in this calculation. The expectation value of x can be written as where li(x) is the Logarithmic integral function [27]. Also, since π a (x) is a slowly varying function of x, we can write the above definite integral as Since from Eq.
/π a ( p n ) and from the prime counting function we can assume π a ( p n+1 ) ≈ π a ( p n ) for sufficiently large n, a formula for the prime gap between two consecutive primes i.e., n th and (n + 1) th prime can be estimated as To check the validity of Eq. (6),we have first calculated the prime gaps using Eq. (6), where the upper and lower limits are taken as the exact primes. As it can be seen in Table 1, the calculated values of prime gap on the basis of Eq. (6) given in the 4th column of the table matches with that of the exact gap values as provided in the 3rd column of the table and thus confirm the validity of the equation for the determination of prime gap. Though π(x) is a discrete valued function, its asymptotic form given by x lnx is a continuous function of variable x. This motivates us to modify Eq. (6) to obtain the prime gaps which does not require the exact primes as the lower and upper limits but instead use the asymptotic form of prime number derived from the Prime number theorem p n ≈ nln(n) with the substitution n = χ where χ is the continuum limit of the prime index n in p n . By using the pair of successive prime numbers through their asymptotic approximation χlnχ , (χ + 1)ln(χ + 1), Eq. (6) appears as, The prime gaps calculated using Eq. (7) is provided in the 5th column of the Table 1. We see that since we are using the approximation for the prime as χlnχ , we only get monotonically increasing gaps. Butg n does not give the correct values due to the fact that the simultaneous use of twin asymptotic forms of PNT like n = p n /lnp n and p n = nlnn introduces an error in the calculation of n by a factor c = (1 + lnlnn lnn ) −1 , which amounts to 26.6% at n = 25 to 6.2% at n = 1.5 × 10 27 . We now propose a formula for the maximal prime gap on the basis of the prime gap obtained in Eq. (7). The maximal prime gap [28] defined as g x = max( p n+1 − p n , p n ≤ χ), can be obtained by substituting π a (χ ) instead of π a (χln(χ )) in the denominator of Eq. (7), which amounts to scaling Eq. (7) with π a (χlnχ) π a (χ ) . This scaling factor infect is equal to c lnχ and with this scaling factor the formula for the maximal prime gap appears as The values of maximal gap computed on the basis of Eq. (8) and tabulated in the 6th column of Table 1 provides a very good estimate when compared with the exact values as given in 3rd column of the table. In the 7th column of Table 1, we have provided the prime gap estimated by Cramer [29]. For a comprehensive understanding, the exact gap, the gap calculated on the basis of Eq. (6), the maximal gap calculated through Eq. (8) and the Cramer's estimate on prime gap are depicted in Fig.1. The datas of Table 1 and the graphs in Fig.1 validate the effectiveness of the formulas Eq. (6), Eq. (8) proposed for the prime gap and the maximal prime gap respectively. In the table, we have taken all the primes up to 113 and after that we have gone up to the prime number 4652507 taking into consideration only the maximal gaps.
The other two tables such as Tables 2 and 3 tabulate some twin primes [30,31], with gap 2 and their calculated gap g n . Table 1 Comparison of the exact gaps [30] with the calculated gaps g n from Eq.(6),g n from Eq. (7), g x from Eq.(8) and the gaps given by Cramer

Prime
Next prime Exact gap g n (Eq. (6))g n (Eq. (7)) g x (Eq. (8) (6))g n (Eq. (7)) g x (Eq. (8) Here in this table we have all the primes up to 113 and after that some primes up to 4652507 which produce the maximal gaps. g n is almost equal to the exact gap.g n is not a good approximation. It is seen that g x is a very good approximation to the maximal gap which increases with increase in prime size  Table 1. The top most line with − − − is the plot obtained by Cramer [29] for the maximal prime gap which is 2 times bigger than the exact values. The solid green line is the plot of g x of Eq. (8). The inset in the plot shows the variation of prime gap size with with prime size in the range of primes up to 113

Classical dynamics
Using the interaction potential Eq. (3), one can study the hidden classical dynamics characterizing prime numbers through the corresponding classical Hamiltonian system. As we will see that this classical study reveals, re-establish certain interesting and intriguing physical and mathematical features such as connection [7] between prime numbers and black holes and also the appearance of elliptic curves [32] used in cryptography through this classical analysis.
The plot of potential V (x) as a function of x is given in Fig. 3. The potential V (x) = 0 at x = e −3/2 = 0.223 and is negative below this value of x up to x = 0. Also one can see that decreases and goes to zero as x → ∞. So there are singularities both at x=0 and 1. However it is worth pointing here that prime numbers exists only for values more that 2 (x ≥ 2). Using the Hamilton's equation of motion [33], we calculate the force experienced by a particle moving in the potential V (x) given by Eq. (9) as Where p represents the linear momentum. Eq. (10) represents the force experienced by the particle at the position x. The force on the particle decreases and tends to zero for x → ∞.

Phase space, trajectories and zeroes
Now we study the classical aspects of the problem in the phase space of x and p by considering the classical dynamics of a single particle corresponding to Eq. (1) in the system of units as mentioned earlier with m = 1, based on the potential function as given in Eq. (9), the classical Hamiltonian for a single particle system can be written as, The potential function in the above Hamiltonian is nonlinear in x hence the nonlinear dynamics [11] of this system need to be studied. In nonlinear dynamics the trajectories and zeroes or fixed points are very important quantities which reveal the interesting dynamics peculiar to the system. Hence we plot in Fig. 4 the relation between x and p for a given energy parameter E using the Eq. (11). To be specific we choose E = −10, 0, 1, 10, 100 and analyze those trajectories on the x-p plane. For E > 0 only two zeros (x 1 ,x 2 ) exits in each graphs which are indicated by circles (x is a zero or a fixed point ifẋ = p = 0 at that point). These two zeroes are half stable and half unstable fixed points. On the right branch of each of the x − p plot with different E of Fig. 4, if a particle has positive velocity, then it will go away from the right side zero to ∞ and if it has negative velocity, it will come to that zero and stay there. On the other hand in the left branch, it is opposite in nature that is if the particle has positive velocity then it will move to the fixed point and if it has negative velocity, then it will go away from it. As is done in standard nonlinear dynamics study, stability is shown in darkness and instability in emptiness of the small circles drawn at the fixed points. We also see that as the value of energy parameter E increases, the two zeroes come closer and when E → ∞ they merge at x = x 1 = x 2 = 1. For E = 0, only the left branch exists which cuts the x-axis at x 2 = 0.223 and x 1 moving to infinity. As E becomes negative, the left side zero occur between 0 and 0.223.

Bifurcation dynamics and the distance between the two zeroes
As discussed in the subsection-B, since the fixed points [11] are important in describing the dynamics, we make a detail study of these fixed points or zeroes and the bifurcation phenomena [11] of the mechanical system. In our case, the qualitative changes in the dynamics occurs when the energy parameter (E) is varied. For E < 0, there is only one saddle node remaining between 0 and 0.223. At E = 0, another saddle node appears from the blue sky and for E > 0, both the saddle node approaches each other. So E = 0 can be considered as bifurcation point where there is a splitting into two branches. This point is made clear in the inset of Fig. 5. This bifurcation seems to be a case of blue sky bifurcation [11].
We then try to find an analytic expression for the distance between the two zeroes as a function of energy E. For this purpose one need to know the positions of the two zeroes as functions of energy E. Putting p = 0 in Eq. (11) Now we substitute x = e y , or (ln(x) = y), and write the above equation as y − 4E y 2 e 2y + 1.5 = 0 As a single closed form analytic expression for y as a function of energy E in the entire region of −∞ < E < ∞ for the above equation is difficult to obtain, we first solve the equation numerically and the plot of y as a function of energy E is given in the Fig. 5. Now we try to find analytic expressions of y as functions of energy E in different regions of energy. We consider three regions of energy such as E > 0, E = 0, and E < 0. It so happens that E > 0 region corresponds to small value of y; E = 0 region corresponds to large y and E < 0 region covers negative values of y.

Case-1 (E>0)
Considering the value of y as small, we expand and approximate e 2y ≈ 1 + 2y. Using this in the Eq. (13) and neglecting cubic term in y, we get the quadratic equation y − 4y 2 E + 1.5 = 0 which has two solutions expressed as, These two solutions for large E are shown in the right hand side of Fig. 5. The subscript s indicate the small value of y in this region. So the distance between the two zeroes as function of E for E > 0 can be written as The separation δx is plotted in Fig. 6; While the dots are the exact values calculated numerically from Eq. (12), the solid line represents the separation obtained from the analytic expression Eq. (16). As seen from the fig, the analytic expression given by Eq. (16) exhibits a very good agreement with the exact value.
We also see that when energy E approaches 0, the separation between the two zeroes x → ∞ and when E → ∞, x → 0. Case-2 (E=0) For E = 0, the Eq. As it has been seen in Fig. 5 for E < 0, zeros occur in the range 0 < x < 0.223 i.e., −∞ < y < −1.5. For E < 0 and small values of x (very near to 0) corresponding to very large negative values of y, we can neglect 1.5 in the Eq. (13) and get 4E ye 2y = 1 The solution of the above Eq. (17) for any real E is given as y = 0.5W n 1 2E , where W n (x) being the famous Lambert W function [12].
So for E < 0, we have y E<0 = 0.5W which gives y E≤0 ≈ − 1.5 at E = 0. Also we find that as E → −∞, y E<0 → −∞ or we can say as E → −∞, x → 0 as shown in the inset in Fig. 5.
In the Fig.(7) we have shown variation of x as a function of E calculated numerically from Eq. (12) as dots and the values obtained from Eq. (18) as a solid line given by which shows reasonably good agreement. The slight variation is due to the approximation used.
Since Eq. (17) is also valid for positive E and large positive y its solution can be expressed in Lambert W function, as such we tried to fit our results for Case-1 and Case-2 with Lambert W functions and we found for 35) where W 0 corresponds to the principal branch of the Lambert W function.

Connection between primes and black holes
The appearance of the Lambert W function in the prime number analysis is deep rooted at different levels of analogy. It has been shown [13] recently that the prime counting function π(x) (x → ∞) is approximately equal to exp(W 0 (x)) where W 0 (x) is the principal branch of the Lambert W function. Also the Lambert W functions (both W 0 and W −1 ) appear in General Relativity in the solution [14,15] for a two body system in one space and one time dimension. Prime numbers and their connection to black hole physics which has been speculated earlier [7]  . Now we discuss the importance of the four acceleration [19] (whose forms are given in appendix-B) in resolving the divergence in the prime potential.
According to General Theory of Relativity, at the Schswarzschild radius, the four acceleration is divergent so that the particle experiences infinite force of attraction at that surface and must enter into the black hole, though the potential and the three forces are finite yet the four force is infinite. In our prime potential case the three force becoming divergent is a boon than a hurdle so that the four force apparently becomes also divergent, which indirectly says that x = 1 is the Schwarzschild radius.
In Fig. 8, we show in a pictorial way, that the analogy appears to be deeply ingrained in the classical description of the two fixed points interpreted as coordinate time t and proper time τ . In the fig, we make a replot of Fig. 5 by scaling y as y/(−0.4) for the lower branch ( the multiplication by negative sign to make the associated proper time τ positive and division by 0.4 to make the corresponding proper time τ to lie below the coordinate time t). The above multiplication by negative sign [7] though looks arbitrary, is not without the justification because centrifugal inversion [7] with a negative orbital number also appears in the calculation of black hole physics. We identify in Fig. 8, y values as coordinate time t and proper time τ , energy as distance from the black hole (E ∼ r ) and Schwarzschild radius at E = 0 where coordinate time t goes to infinity and proper-time τ is finite. Also we see that t and τ both become equal at x → ∞ from the black hole. The inset in Fig. 8 is the actual diagram [34] for a black hole as calculated in General Theory of Relativity.In the actual diagram the time synchronization (that is (r ) is structurally almost similar to the general relativistic potential V GTR eff (r ), both having maximum, minimum and same behaviour as r → +∞ or r → 0, except for the fact that the maximum is a positive divergence in case of V Prime eff (r ). According to General Theory of Relativity, at the Schswarzschild surface, the four acceleration is divergent so that the particle experiences infinite force of attraction at that surface and must enter into the black hole, though the potential and the three force is finite yet the four force is infinite. In our prime potential case the three force becoming divergent is a boon than a hurdle so that the four force apparently becomes also divergent, which indirectly says that x = 1 is the Schwarzschild radius In this section we derive an expression for the upper limits of prime gaps using the inverse stereographic projection [17,18] which relates the points on the real line to the points on a unit circle (Fig. 10) and hence the gaps on the real line to the projected arcs on the unit circle. Though the calculation of prime gaps from the gaps on the unit circle is a purely geometric one, we make an attempt in Sec-IV(C) to analyse quantum mechanically where in the Schrodinger equation is considered in a circular coordinate of radius 1; effectively making it a single coordinate problem involving angular coordinates.
Here though the motion on the real line is transformed to a motion on a circle, the dynamics and the trajectories do not play any role in establishing the upper limits of the prime gaps. They display the periodic motion on the unit circle associated with the previous phase space dynamics by relating the classical trajectories (Fig. 4) to the trajectories (Fig. 11) on the unit circle.
For a given energy 0 < E < ∞,the inverse stereographic projection of the points on the x axis with (x > 1) for the upper and lower part of a trajectory of Fig. 4, exhibits back and forth motion of a particle on a segment of a curve from the equator to the north pole on the circle of projection in Fig. 11a. As energy E increases, the segment size increases and this segment is one quarter of the full circle for points on the x-axis from x = 1 + , to x → ∞ as shown in Fig. 11a and another quarter for −∞ < E < ∞ for the points from x = 0 to 1 as shown in the Fig. 11b.
The other half of the circle is related to x = 0, to x → −∞. Specifically we note that at E = 0 (looking at the Fig. 4), we have one fixed point at x → ∞ and one at 0.223. So we denote them on the unit circle with big dots, one at North pole N in Fig.11a and one on the unit circle away from the south pole P on the line joining N and 0.223 in Fig. 11b. The point corresponding to E → −∞ lies at P denoted by a big dot.
(a) (b) Fig. 11 Plot of trajectories of Fig. 4 when projected onto the unit circle. a is for x > 1 and 0 < E < ∞. b is for 0 < x < 1 and −∞ < E < ∞

Stereographic projection and Prime gap using geometry only
Specifically, stereographic projection ( Fig. 10) from the north pole (0,1) onto the x-axis gives a one-to-one correspondence between the points (x c , y c ) on the unit circle (with y c = 1) and the points on x-axis. If (m/l , 0) is a rational point on the x-axis, then its inverse stereographic projection [17,18] is the point (x c , y c ) given as Since m and l are any integer such that m/l is rational, and as we are concerned with the integers which are primes, we can take m = sth prime ≈ (s ln s) and l = 1. As the point on x-axis (number line) moves towards very large values, the projected point on the unit circle approaches the north pole with x c → 0 and y c → 1. So the two points at very large distances on the x-axis having a separation of will be projected as two nearby points very close to the north pole.
So in order to estimate the prime gap we first calculate the straight line distance between these two projected points on the unit circle. Two points on the x-axis located at a very large distances (x → ∞) subtend a very small arc-length on the projected circle and can be approximated by the chord length. If we choose m 1 = s lns , l 1 = 1 and m 2 = (s + 1) ln(s + 1), l 2 = 1 for the two primes on the real line, we can calculate, using Eq. (19), corresponding coordinates on the unit circle as (x c1 , y c1 ) and (x c2 , y c2 ) respectively and hence the distance between them on the unit circle as Thus the prime gap can be calculated as the reciprocal of g s as In the Table 4 we tabulate the above calculated gapg s and the reported upper limits such as g L D s (Lu and Deng [35]) and d N G s ( Nagura et.al. [36]) for comparison. We plot the above calculated gaps and the exact gaps as function of prime position in Fig. (12). Lu and Deng [35] have recently shown that the prime gap of a prime number p s is less than or equal to the prime count of the prime number that is g L D s ≤ π( p s ) = s for s ≥ 1. Our result given in Eq. (21) thatg s ≈ 0.5 s provide a better upper limit than that of Lu and Deng, though albeit ours is valid for s ≥ 12. Also in the same Fig.(12), we have plotted the upper bound d N G s = (1/5) p s for s ≥ 10 as derived by  Nagura [36] and discussed by Ribenboim [37]. It may be seen from the table that Nagura's upper bound for s upto 30th prime number though remain less, for higher values of s they deviate than the LD approximation and far away from the exact value. Our value for the upper bound remains much smaller than both the previous reported one.

Hamiltonian in θ coordinate on the unit circle and the elliptic curves
Since due to the stereographic projection motion of a particle on the real line can be mapped onto the unit circle, here we express our classical Hamiltonian given by Eq. (11) in terms of the angle coordinate θ to illustrate certain interesting mathematical properties related to the prime numbers. One can parameterise (m/l) = x in Eq. (19) and write where (x c , y c ) are the coordinates on the unit circle corresponding to points on the x-axis with coordinates (x and 0). Using Eq. (22), it is easy to find that x = tan π 4 + θ 2 . The speed of the particle on x-axis can be related to speed on the unit circle asẋ = 1 2 sec 2 π 4 + θ and momentum on the unit circle as It is interesting to note that p(θ ) =θ as a function of θ , plotted in upper panel of Fig. 13 for different values of energies E (which were used in Fig. 4 ) has similarity with the elliptic curves [32] given by y 2 = x 3 +ax +b = f (x) and plotted in the lower panel. The similarities are in the sense "both the curves have genus 1 and though, maximum of three points on an elliptic curve may be on a straight line, there may be maximum of five points on the present figure which are collinear. Also the curve given by Eq. (24) is as symmetric as that given by the elliptic curve shown in the Fig. 13 and each having three distinct roots but our curve has a cusp at θ = π/2". It is interesting to note that recently Beshaj, Shaska and Zhupa [38] had formulated and shown that the elliptic curve is a particular realization (n=2) of the more general superelliptic curves given by y n = f (x). The elliptic curves are related [32] to primality test and are heavily used in communication cryptography.

Quantum dynamics on the unit circle
In this section, a quantum mechanical analysis on the unit circle is given using the classical momentum p(θ ) given by Eq. (24). Taking the square of p(θ ) in Eq. (24) and rewriting we get Here we note that the original potential given by Eq. (9) is modified as − E 4cos 4 ( π 4 + θ 2 ) with the E of Eq. (1) appearing as a parameter in the new θ dependent potential.
This is the Schrodinger equation corresponding to Eq. (1), when particle position on the x-axis is projected to the position of a particle on the unit circle. Here we note again that the original potential is modified and the E of Eq.
(1) appears as a parameter in the new θ dependent potential with eigen value of the above equation as zero. Near θ → 0 (x → 1) the wave function appears as ψ(θ) = Aθ 3/2 . This corresponds to the density ρ(θ) = A 2 θ 3 → 0, which means projected points representing primes onto unit circle become less crowded near the point 1 and more crowded on the right side of the north pole. As we are concerned with the gaps for primes, the prime gaps are equal to the inverse of small arcs between the region 0 to π/2 and hence one can define the expectation value Here θ p n is the angle subtended by the position of the point on the unit circle corresponding to the n th prime p n . When the primes are projected, these unique points (which are not them selves primes) represent the position of primes on the unit circle. For larger primes, the larger prime gaps on x-axis are represented by smaller and smaller arcs on the unit circle towards the north pole. This fact may be useful, since determining large primes and prime gaps are computationally demanding.

Conclusion
In this paper we show that since finding large prime numbers remains computationally very difficult, one can construct a single particle Schrodinger equation from an interacting quantum many-particle system representing the prime numbers through the asymptotic form of the prime counting function and study certain properties of primes. The prime gap (and specifically the maximal prime gap) is constructed as the expectation value of position x over the single particle wave function which shows that the maximal prime gap increases as we go away on the x-axis. The maximal prime gap is estimated by us is better than the previously reported results. After we determine the interaction potential, we study the nonlinear classical dynamics such as the trajectories associated with the above one particle problem and find two fixed points which are half stable and half unstable in nature. The saddle nodes found to undergo blue sky bifurcation with the bifurcation point E = 0. The Lambert W functions appear as solutions of the fixed points as function of energy. Also the stereographic projection which reveals the underling dynamics on the unit circle provides an upper limit on the prime gap.The conjectured connection as proposed by many authors between prime numbers and black hole which is of much transcendence appears serendipitously in the solutions of the fixed points as a simple mathematical identification with different times associated with a black hole. Thus the equivalent physical problem of prime number reveals that though numbers are just points on the x-axis, yet it is possible to extract certain facts about them once we know the resulting equivalent interaction potential that represents them and gives rise to their distribution on the x-axis.

Conflict of interest
The authors have replied that they have no conflicts of interest to declare.

A. Potentials
In order to compare the characteristics of prime potential with the effective potential associated with General Theory of Relativity we write the effective prime potential in 3-D as [22] V Prime eff (r ) = 1 4r 2 1 ln(r ) In the above equation we have taken in the last term −L 2 instead of +L 2 , since it has been conjectured [7] that centrifugal force reversal [39] happens for r < 3M black holes. Also the negative sign is due to quantum anti-centrifugal force being metric related [40] and the same negative sign appear [41] in case of an inverted harmonic half-oscillator with negative fractional phase shift associated with an attractive centrifugal barrier(i.e. well) in relation to prime numbers.
The effective potential in General Theory of Relativity is given as It is to be noted that the mathematical dependence on r though appears to be different in both cases, the graphically they appear to be same.

Appendix B
A. Four acceleration for GTR and prime potential The magnitude of four acceleration defined as a 0 = (−a μ a μ ) 1/2 is given with c = 1 as [19] a 0 = (−a μ a μ ) 1/2 = G M So we see that at r = r sch = 2G M/c 2 for a black hole the magnitude of the four acceleration diverges. But since the four acceleration has to be divergent in GTR for a black hole , our potential having a divergence is a boon since it mimics the four acceleration for a black hole and defines the Schwarzschild radius at r = 1.