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Can someone provide a proof for the following claim?

$$\pi=\dfrac{S_0S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \pmod{4} } \frac{p}{p+1}\right)$$

where $M_3=2^3-1$ ; $M_5=2^5-1$ : $p$ -exponent of Mersenne prime ; $S_2$ the third term of the Lucas sequence $S_i=S_{i-1}^2-2$ with $S_0=4$ , that is $S_2=194$ .

For the first $50$ odd exponents of Mersenne primes I get value: $3.141592655591489672\ldots$ while the real value of $\pi$ rounded to 18 decimal digits is $3.141592653589793238$ . You can find a list of exponents of Mersenne primes here.

Pari/GP code.

Here is graph and absolute differences between value of $\pi$ and values of formula:

graph.pgn

[0.45957, 0.21094, 0.20812, 0.036331, 0.23495, 0.066124, 0.034117, 0.017674, 0.053575, 0.023990, 0.00074136, 0.0052987, 0.00012293, 0.0023315, 0.0037559, 0.0023796, 0.0014035, 0.00066499, 0.0013750, 0.0010508, 0.00073488, 0.00045475, 0.00029719, 0.00015243, 1.7066 E-5, 5.3538 E-5, 1.7110 E-5, 1.1319 E-5, 1.2472 E-5, 2.0665 E-6, 6.2174 E-6, 2.5620 E-6, 5.0597 E-6, 2.8129 E-6, 1.7574 E-6, 7.1759 E-7, 2.6702 E-7, 3.3743 E-8, 1.8337 E-7, 3.1407 E-7, 4.3506 E-7, 3.3173 E-7, 2.3531 E-7, 3.1986 E-7, 2.4619 E-7, 1.7332 E-7, 1.1905 E-7, 7.6714 E-8, 3.6037 E-8, 2.0017 E-9]

EDIT

Let's introduce the following notation $L_n^{(1)}(P,Q)$ - Lucas number of the first kind and $L_n^{(2)}(P,Q)$ - Lucas number of the second kind. Note that $S_0=L_1^{(2)}(4,1)$ , $S_2=L_4^{(2)}(4,1)$ , $M_3=L_3^{(1)}(3,2)$ and $M_5=L_5^{(1)}(3,2)$ . So we can rewrite expression above as follows: $$\pi=\dfrac{L_1^{(2)}(4,1)L_4^{(2)}(4,1)}{L_3^{(1)}(3,2)L_5^{(1)}(3,2)} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} \atop p \in \mathbb{M} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \pmod{4} \atop p \in \mathbb{M}} \frac{p}{p+1}\right)$$ where $\mathbb{M}=\{p | p \in \mathbb{P} \text{ and } 2^p-1 \in \mathbb{P} \}$

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    $\begingroup$ OEIS A000043 notes that "It is believed (but unproved) that this sequence is infinite." A proof of your claim would also be a proof that there are infinitely many Mersenne primes, since otherwise $\pi$ must be rational. Therefore either the answer is "No, no-one can provide a proof" or OEIS needs to be updated. $\endgroup$
    – Peter Taylor
    Commented Jul 2 at 14:15
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    $\begingroup$ $\frac{776}{217}$ is quite a weird fraction, and the accuracy isn't very high, so I believe this is likely just a numerical coincidence. Do you have any non-numerical reason to believe this? $\endgroup$
    – Daniel Weber
    Commented Jul 2 at 14:45
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    $\begingroup$ @Pedja In that case, this would be absolutely shocking if true. $\endgroup$
    – JoshuaZ
    Commented Jul 2 at 16:13
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    $\begingroup$ @Pedja The value you are actually getting is $0.8785123\dots$. While you're saying it's approximately $\frac{217}{776} \pi$, it could just as well be $\frac{1063}{1210}$, or $\frac{11 + 30\log(2) + \frac{2}{\log(3)} - 25\log(3)}7$, or just some arbitrary value which doesn't have a closed form. $\endgroup$
    – Daniel Weber
    Commented Jul 2 at 16:23
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    $\begingroup$ @TheSimpliFire Very likely the product was one similar to this but over all primes 1 mod 4 in one term and all primes 3 mod 4 in the other term. In that case, they can be special values of L functions and related. But dropping out the ones that are not Mersenne exponents is going to be highly irregular in terms of what it does to the product. $\endgroup$
    – JoshuaZ
    Commented Jul 2 at 22:28

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