FIND PRIMES USING EULER'S FORMULA

James Constant

math@coolissues.com

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 o_{n} 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 o_{n}~*n/lnn* and saying that o_{n} 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*

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 *p _{n},*
in effect confirming the PNT.

** **If prime number *p _{2}*
is greater than a previous prime number

In what follows, for purposes of
illustration and simplicity, I will make several assumptions. My first
assumption is that *(p _{1}p_{2})^{x}>>p_{1}^{x}+p_{2}^{x}.*Accordingly, equation (2) can be rewritten as

Since the separation between primes is

* p _{2}=p_{1}+2k
k = 1,2,3, . . .*
(5)

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

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** **

in which series *S _{k}*
is an irrational number. This, however, is not possible in view of
equation (5). Since

** **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<<p _{1}/x*,

* p _{2}=p_{1}(1+2k/p_{1})
k = 1,2,3, . . .*
(11)

there are a finite number 2*k*
of locations between primes *p _{2}*
and

** ** The PNT states that the *nth*
prime *p _{n}* is of the order (~)
of

** **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<<p _{1}/x*
and, fourth, the exact values of play
no role in finding primes in the region

** **The foregoing results are equally
applicable to Riemann's conjecture which, if true, must demonstrate
that *p _{2}=p_{1}S_{k}*
in which

^{1}** ****"Analytic Number Theory: The Prime Number
Theorem." Britannica CD, Version 99© 1994-1998.
Encyclopædia Britannica, Inc. Retrieved from **__http://users.forthnet.gr/ath/kimon/PNT/Prime%20Number%20Theorem.htm__

^{2}*See* Riemann's
Analytic Extension Disproved http://www.coolissues.com/mathematics/Riemann/disproof.htm

^{3}*See* plot of Riemann zeta function for
real s>1 at http://en.wikipedia.org/wiki/Riemann_zeta_function

**Copyright 2009 by James Constant**

**By the same author:****
http://www.coolissues.com/mathematics/sameauthor.htm**