Please review and comment

Please review the following and the proposed theorem T1. If there are
errors I would appreciate the correction. If there are no errors, I
would appreciate your professional suggestion as to the best approaches
to either prove or disprove T1.

-- begin --

Notations & Definitions:

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[N1] n-th Prime:
p_n is the n-th prime, (where p_1 = 2, p_2 = 3, etc.)


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[N2] n-th Primorial, p_n#:
p_n# = PRODUCT of p_1 * p_2 * p_3 * ... * p_n


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[D1-a]: Congruence System D2 [for Delta-2] for p_n#

D2(n):
{
x == 1 Mod 2
x == 2 Mod 3
..
x == r_i Mod p_i
..
}
with constraints:
(i) 3 < i =< n
(ii) 0 < r_i < p_i
(iii) r_i != p_i-2


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[D1-b]: Function Phi'(D2, n) as the number of distinct solutions of
D2(n) for x, where 0 < x < p_n#. (obviously x=1 is always a solution.)

Phi'(D2, n) = PRODUCT (p_i - 2) where 3 < i =< n

ex:
Phi'(D2, 4) = 3 [p_4 = 5, p_4# = 30]
Phi'(D2, 5) = 15 [p_5 = 7, p_5# = 210]
Phi'(D2, 6) = 135 [p_6 = 11, p_6# = 2310]
etc..

In modular form, we see that Phi'(D2, n) is the precise number of
residue pairs (r1, r2) Mod p_n#, where r2 == r1-2 Mod p_n#.

Effectively, Phi'(D2, n) - 1 provides the number of delta-2 pairs
*relatively* prime to all primes up to p_n in the range 0 ... p_n#.
(We are excluding the ever present (1, p_n# - 1) Mod p_n#)

ex:
Phi'(D2, 4) = 3 [p_4 = 5, p_4# = 30]
[(1, 29)] (11, 13), (17, 19)

etc..

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[N3]
Let p_w denote the next prime after p_n, i.e. p_w is the (n+1)-th
prime. (Simply to avoid ambiguous notation p_n+1.)


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Axiom [A1]:
It is assumed as an axiom that p_w << p_n#.

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Lemma [L1]:

For any solution p' of D2(n), in the range p_w < p' < p_w^2, the pairs
(p'-2, p') are both absolutely prime. Where p' = p_w+2, the prime
pairs are (p_w, p').

Clearly if p' is a solution to D2(n), it is relatively prime to all
primes up to and including the n-th prime p_n.

Further, for any k within the specified range [e.g. p_w < k < p_w^2],
the number k is either itself a prime greater than p_w, or, a k is a
product of one or more primes less than p_w.

Thus, if p' is a solution to D2(n), then p'-2 and p' must both be
prime.


Proposed theorem [T1]:

For any n, there exists at least one solution p' to D2(n).

-- end --

I believe axiom [A1] is not subject to dispute.

[T1] then is clearly a specialization of the twin-prime theorem:

[T1] states that for any prime p_n, there exists at least one twin
prime pair of form (p'-2, p') in the range 0 < p' < p_n#. (p' is
clearly at least the 2nd subsequent prime after p_n.)

[Q1]: Is it possible to dis-prove T1?

[Q2]: Is it possible to prove T1?

[Q1] is answered 'Yes' does not immediately disprove the twin-prime
problem, but it *may* be possible to show that if [T1] does not hold
for ALL n, then perhaps some contradictions arise in conjunction with
other proven theorems in NT which assume the twin prime. (I have no
input as to what these may be. It is just a notion.)

If [Q2] is answered 'Yes', then we have effectively proven the
twin-prime problem. It affirms that there are an infinite number of
primes of form (p-2, p).

if both [Q1] and [Q2] are answered 'No', then again the intuitive
notion is that perhaps this may help towards proving the Twin-Prime
problem as undecidable.

As with the Twin-Prime conjecture itself, the evidence seems
overwhelming that [T1] is true. The assumption here is that perhaps T1
is *easier* to prove than the general TP.

Your corrections and feedback are appreciated.

/R

Joubin Houshyar

.

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