## FW: The MAGIC KEY of primes PROOF

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• From: dim-val@hotmail.com To: dim-val@hotmail.com; kon_val@hotmail.com Subject: The MAGIC KEY of primes PROOF Date: Sun, 24 Jun 2012 03:15:55 +0300 Hi Let the
Message 1 of 1 , Jun 25, 2012
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From: dim-val@...
To: dim-val@...; kon_val@...
Subject: The MAGIC KEY of primes PROOF
Date: Sun, 24 Jun 2012 03:15:55 +0300

Hi

Let the
pairs(e1, e2)wheree1, e2are even numbers, e2-e1 = 2
andp = (e1 + e2) / 2is a primenumberamong
them?
andapply therulein
(4,5,6), (6,7,8), (​​10,11,12), (12,13,14),(16,17,18), (18,19,20), (22,23,24), (​​28,29,30),
(30,31,32),(36,37,38), (40,41,42), (42,43,44), (​​46,47,48),
..., (e1, p, e2)
The rule is:
ifp = 3
mod 4(ore1 = 2
mod 4)then  we formthe
productΠe2/e1
ifp
= 1 mod 4(ore1 = 0 mod 4) then  we formthe productΠe1/e2

The product of:
(4/6) * (8/6) * (12/10) * (12/14) *
(16/18) * (20/18) * (24/22) * (28/30) * (32/30) * (36/38) *
(40/42) * (44/42) * (48/49)* ... = 1
for each p prime and  p>
3, p -> {5,7,11,13,17,19,23, ..}

PROOF:

Π(1 -X(Pk)/Pk)
= 4/pi for k=2 to infinite, Pk is kth prime.
Π(1+X(Pk)/Pk)
= 2/pi for k=2 to infinite, Pk is kth prime.

Where:
{ +1  if P=1 mod 4
X(P)={
{ -1   if P=3 mod 4

For k>=3 the 2
products are equals.
Π(1 -X(Pk)/Pk) = 3/pi
Π(1+X(Pk)/Pk) = 3/pi

Because for k=2 we
have (1+1/3)=4/3 and (1-1/3)=2/3
So 4/pi*3/4=3/pi
and 2/pi*3/2=3/pi

The quotient
(Π(1 -X(Pk)/Pk)) / (Π(1+X(Pk)/Pk)) = ((1-1/5)/(1+1/5))*((1+1/7)/(1-1/7))*…=
=(4/5)/(6/5)*(8/7)/(6/7)*… = 4/6*8/7*12/10*… = (3/pi)/(3/pi)=1  for k>=3

(I call this the MAGIC KEY of primes.)

Best regards
Dimitris Valianatos

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