I would like to

give you more information about my conjectures for prime numbers.

I will give you

2 conjectures for prime numbers, which gave me the idea

for the twin primes conjecture.

(These 2 conjectures I was able to prove, so now are two important theorems

in the theory of prime numbers. I will send you the proof.)

C)

First see the

correlation with the twin primes.

As we saw the twin primes are pairs (p1, p2) where p1, p2 are primes p2-p1 = 2

and e = (p1+p2) / 2 is an even number among them.

Thus we have

the triples: (3,4,5), (5,6,7), (11,12,13), (17,18,19), (29,30,31),

(41,42,43),

(59,60,61), (71,72,73), ..., (p1, e, p2)

e -> {4,6,12,18,30,42,60,72, ...} (some numbers e divisible by 3 and some with

3 and 4, ie 12).

and apply the rule.

if e = 0 mod 4 or

p1 = 3 mod 4 then we form the product Πp2/p1

if e = 2 mod 4 or p1 = 1 mod 4 then we form the product Πp1/p2

So my

conjecture says that the product:

(3 ^ 2 / 2 ^

2) * (5 ^ 2 / 3 ^ 2) * (5 ^ 2 / 7 ^ 2) * (13 ^ 2 / 11 ^ 2) * (17 ^

2 / 19 ^ 2) * (41 ^ 2 / 43 ^ 2) *(61 ^ 2 / 59 ^ 2) * (73 ^ 2 / 71 ^ 2) * (101 ^

2 / 103 ^ 2) * ... = pi?

3.1887755102040816321

to 1e 1 (3 ^ 2 / 2 ^ 2 * 5 ^ 2 / 3 ^ 2 * 5 ^ 2 / 7 ^ 2)

3.2055606708805624550 to 1e 2

3.1290622219773513145 to 1e 3

3.1364540609918890779 to 1e 4

3.1384537326021492746 to 1e 5

3.1417076006640026373 to 1e 6

3.1417823471756806475 to 1e 7

3.1415377533170544536 to 1e 8

3.1415215264211035597 to 1e 9

3.1415248453830039795 to 1e 10

3.1415126339547108140 to 1e 11

3.1415144504088659201 to 1e 12

3.1415142045284687040 to 1e 13

3.1415144719058962626 to 1e 14

3.1415384423175311229 to 1e 15

A)

What happens if

we take all pairs (e1, e2) where e1, e2 are even numbers, e2-e1 = 2

and p = (e1 + e2) / 2 is a prime number among them?

and apply the rule in the following triads?

(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:

if p = 3 mod 4 (or

e1 = 2 mod 4) then we form the product Πe2/e1

if p = 1 mod 4 (or e1 = 0 mod 4) then we form the product Πe1/e2

B)

Similarly, what would happen if we take all pairs (e1, e2) where e1, e2 are evens,

e2-e1 = 2

and c = (e1 + e2) / 2 is odd composite number among them?

(8,9,10), (14,15,16), (20,21,22), (24,25,26), (26,27,28), (32,33,34), (34,

35,36), (38,39,40),

(44,45,46),

..., (e1, c, e2)

c -> {9,15,21,25,27,33,35,39,45, ...}

and apply the rule

in the above triads?

The rule is:

if c = 3 mod 4 (or

e1 = 2 mod 4) then we form the product Πe2/e1

if c = 1 mod 4 (or e1 = 0 mod 4) then we form the product Πe1/e2

We start with the last (B).

The first conjecture

says that:

The product 4 * (8/10) * (16/14) * (20/21) * (24/26) * (28/27) * (32/34)

* (36/34) * (40/38 ) *

* (44/46) * ... = pi = 3.1415926 ....

But as in the

case of twin primes, the product is unstable

and palindromic around piand the convergence is slow.

Some analytical results.

odd composite

numbers

UBASIC program

10 word -10: point -10

20 P1 # = 1: T = 10

30 for N = 5 to 1000000000 step 2

35 if N> T then T = T * 10: print P1 # * 4

40 if prmdiv (N) = N then 60

50 if N @ 4 = 1 then P1 # = P1 # * (N-1) / (N +1) else P1 # =

P1 # * (N +1) / (N-1)

55 'if (N-1) @ 4 = 0 then P1 # = P1 # * (N-1) / (N +1) else P1 # =

P1 # * (N +1) / (N-1)

60

next

70 print P1 # * 4

3.199999999999999999999999999999999999999999999997 to

1e1 4 * 8/10 (8,9,10)

3.279415297767899415305993497227971481399990346715 to 1e2

3.155979662006539119713526019131092158956361784036 to 1e3

3.150151075329605914570960459016508173589885261865 to 1e4

3.143746673647491490494610656405831223203561991418 to 1e5

3.141609524070628735014614395082987758435483128906 to 1e6

3.14170482178770497125598436238195628437724160566 to 1e7

3.141634736105752586243236584305600545831356879616 to 1e8

3.14160924841725664885331621543307061501208415177 to 1e9

You see the similarities with the conjectureof twin primes.

Thus began my idea for twin primes.

The second conjecture concerns

the (A) case and completes the first conjecture.

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, ..}

I believe that,

this formula is important.

This is the MAGIC KEY of primes.

odd prime numbers

UBASIC program

10 word -10: point -10

20 P1 # = 1: T = 10

30 for N = 5 to 1000000000 step 2

35 if N> T then T = T * 10: print P1 #

40 if prmdiv (N) <> N then 60

50 if N @ 4 = 1 then P1 # = P1 # * (N-1) / (N +1) else P1 # =

P1 # * (N +1) / (N-1)

55 'if (N-1) @ 4 = 0 then P1 # = P1 # * (N-1) / (N +1) else P1 # =

P1 # * (N +1) / (N-1

60

next

70 print P1 #

0.888888888888888888888888888888888888888888888888

to 1e1 4/6 * 8/6 (4,5,6),(6,7,8)

0.96760055745282143922221828990985483378601270138 to 1e2

0.996437288023449634020415265255177534675786722749 to 1e3

0.99738290432080346774903552857486476795753303024 to 1e4

0.999324817106964420775297704764046776745247682675 to 1e5

0.999995629983132555796907941029579460478280788274 to 1e6

0.999964397024871668804107586546482512211561065927 to 1e7

0.999986614898431897369394965700537619210518140771 to 1e8

0.999994718730256121405086897913065623057935239262 to 1e9

The product is

unstable and palindromic around 1and the convergence is slow.

The proof of

the 2 above conjectures are easy.

From all the

above, we arrive in lots of conclusions.

Lemma 1

Another proof

that prime numbers are infinite.

Chebyshev's bias

The above magic

key, perhaps answer the question why

that primes congruent

to 3 modulo 4 seem to predominate over those congruent to 1

Prime number

theorem

Inverse of RH

I'll detail in

the next e-mail, because I have difficulty with English.

(I translate slowly)

D)

{An even case

(D) that required more research to complete the cycle is the following:

We take all

pairs of “twin” nonprimes odd numbers (c1, c2) where c1, c2 are pairs of

consecutive nonprime odd numbers c2-c1 = 2

and e = (c1+c2)

/ 2 is an even number among them.

Thus we have

the triples: (25,26,27), (33,34,35), (49,50,51), (55,56,57), (63,64,65),

(75,76,77), (85,86,87), (91,92,93),

(93,94,95),

(115,116,117), (117,118,119), (119,120,121), (121,122,123), (123,124,125),

(133,134,135), (141,142,143),

(143,144,145),

(145,146,147), ..., (c1, e, c2)

e -> {26,34,50,56,64,76,86,92,94,116,118,120,122,124,134,142,144,146,

...,e,…}

(some numbers e

divisible by 3 and some with 3 and 4, ie 12).

and apply the rule.

if e = 0 mod 4

(or c1 = 3 mod 4) then we form the product Πc2/c1

if e = 2 mod 4 (or c1 = 1 mod 4) then we form the product Πc1/c2

The following product

is likely to converge at???

2*sqrt(2)/pi = 0.9003163161571060695… ???

(25/27)*(33/35)*(49/51*(57/56)*(65/64)*(77/75)*(85/87)*(93/91)*(93/95)*(117/119)*…=?

2*sqrt(2)/pi.}

Best regards

Dimitris

Valianatos

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