James Constant

        Since antiquity, mathematicians have labored to find and apply prime numbers. These numbers extend from 1 to infinity. The Prime Number Theorem (PNT) tells us that the number of primes on in the number interval n is approximately n/lnn. The PNT was first conjectured by Gauss and Legendre around 1800 and was proven by Hadamard and independently by Poussin in 1859. The PNT is written simply as on~n/lnn and saying that on is asymptotic to n/lnn as .1

        Proof of the PNT was one of the great achievements of analytic number theory. The next step was to reduce or eliminate its approximation. Riemann attacked the problem using a formula devised in 1737 by Euler which states that for every real x>1

                    x>1                                                                (1)

where the product runs through all primes.Euler's sum and product (1) converge absolutely for all values of x>1. Since e(x)>1, therefore, it has no zeros in the region x>1. This means there are no exact values for primes pn, in effect confirming the PNT. To obtain zeroes and finding primes, Riemann conjectured that Euler's formula could be extended to the region 0>x>1.2 Using Euler's formula and the binomial series I find a way of efficiently approximating the problem of finding primes without zeroes in region x>1.

        If prime number p2 is greater than a previous prime number p1then

                    p2>p1                                        (2)

        In what follows, for purposes of illustration and simplicity, I will make several assumptions. My first assumption is that (p1p2)x>>p1x+p2x.Accordingly, equation (2) can be rewritten as

                                                  p2>p1                                           (3)

and then solved for


        Since the separation between primes is

        p2=p1+2k                 k = 1,2,3, . . .                                                                 (5)

we can combine equations (4) and (5) to obtain


        My second assumption is that . Accordingly, equation (6) can be written as




from which I conclude that is a rational number and must be integers. The same result can be obtained by reversing signs in equation (3) in which case


        My third assumption is that that . Accordingly, equation (4) can be expressed as a converging binomial series



in which series Sk is an irrational number. This, however, is not possible in view of equation (5). Since p2 in equation (5) is an integer, equation (10) and my third assumption that are impossibilities. When , my second assumption, equation (8) holds and equation (10) approximates equation (5).

        Note that Euler's formula (1) quickly approaches unity, especially when x>10.3 Thus, with x>10 in equations (8) and (10), varies extremely slowly just above zero as . Since 2k<<p1/x, p1>20x. The region 200<20x<p1, therefore, is necessary but not sufficient to find prime numbers p2>p1. Sufficiency is provided by setting conditions 2k<<p1/x,200<20x<p1. In practice this means neglecting low end primes.

        From equation (5) written as

        p2=p1(1+2k/p1)                 k = 1,2,3, . . .                                                          (11)

there are a finite number 2k of locations between primes p2 and p1. Since odd numbered prime p2 cannot be a product of integers, to find primes p2 the term (1+2k/p1) cannot be a prime or product of primes and when multiplied by p1 must be an odd number for each value of k. Each odd number must be tested for primality until prime p2 is obtained. Computationally, one finds (1+2k/p1) and multiplies by p1 and then finds the primality of p1(1+2k/p1) for each value of k. Elsewhere, I show that 2k~lnp1 thus reducing the number of computations. See Finding Prime Numbers at

        The PNT states that the nth prime pn is of the order (~) of nlogn and that the number of primes in interval n is of the order of n/lnn. More accurately, equation (11) tells us that the nth prime pn is precisely p2 and the number of primes in the interval p2= number of primes before p2 + 1 (p2).

        A reader might object to using Euler's equation (1) as a starting point to arrive at the result of inherently obvious equation (11). However, the point is that result equation (10) obtained using Euler's formula equation (1) to find primes is an impossibility except when my second assumption occurs in which case result equation (10) approximates equation (11). The process of going through equations (1)-(10) shows, first, that finding primes in the region x>1 is possible, second, Euler's equation (1), for the purpose of finding primes in the region x>1, is valid only for condition (8) and thus are integers, third, there are upper limits to the distance or gap between primes 2k<<p1/x and, fourth, the exact values of play no role in finding primes in the region x>1.

        The foregoing results are equally applicable to Riemann's conjecture which, if true, must demonstrate that p2=p1Sk in which Sk is a function ofk obtained using Euler's formula, as it appears in Riemann's zeta function, and that this result equals or at least approximates equation (5). According to the foregoing analysis, the reconciliation of Riemann's conjecture with equation (5) is an impossibility.

1 "Analytic Number Theory: The Prime Number Theorem." Britannica CD, Version 99¬© 1994-1998. Encyclop√¶dia Britannica, Inc. Retrieved from 

2 See Riemann's Analytic Extension Disproved

3See plot of Riemann zeta function for real s>1 at

Copyright 2009 by James Constant

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