Re: Please review and comment

Please note correction of constraint (i) for [D1-a]:

(i) 3 =< i =< n

Joubin Houshyar wrote:
> 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:
>
> ///////////
>
> [N1] n-th Prime:
> p_n is the n-th prime, (where p_1 = 2, p_2 = 3, etc.)
>
>
> ///////////
>
> [N2] n-th Primorial, p_n#:
> p_n# = PRODUCT of p_1 * p_2 * p_3 * ... * p_n
>
>
> ///////////
>
> [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
>
>
> ///////////
>
> [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..
>
> ///////////
>
> [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.)
>
>
> ///////////
>
> Axiom [A1]:
> It is assumed as an axiom that p_w << p_n#.
>
> ///////////
>
> 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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