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Note: I don't know how to program a computer to check for me, hence my question. Thanks in advance.

Building on the concept explored in Sets of Prime and Composite Numbers

Conjecture: For every pair of twin primes $a$ and $b$, performing the arithmetic operation of:

$(a+b)+(a ⋅b)= c$

[Example: $(3+5)+(3⋅5)=23$]

yields either a composite number part of a Twin-Composite pair or a prime only within a Prime-Composite pair.

[Example: Twin Composites, Example: 119,121 (composite followed by a composite)]

[Example: Prime-Composite, Example: $23,25$ (prime followed by a composite)]

First Ten Examples from Manual Calculation:

$a = 3$ and $b = 5$ ; yields $23$ ; $23$ is prime, $25$ is composite

$a = 5$ and $b = 7$ ; yields $47$ ; $47$ is prime, $49$ is composite

$a = 11$ and $b = 13$ ; yields $167$ ; $167$ is prime, $169$ is composite

$a = 17$ and $b = 19$ ; yields $359$ ; $359$ is prime, $361$ is composite

$a = 29$ and $b = 31$ ; yields $959$ ; $959$ is composite, $961$ is composite

$a = 41$ and $b = 43$ ; yields $1847$ ; $1847$ is prime, $1849$ is composite

$a = 59$ and $b = 61$ ; yields $3719$ ; $3719$ is prime, $3721$ is composite

$a = 71$ and $b = 73$ ; yields $5327 $ ; $5327$ is composite, $5329$ is composite

$a = 101$ and $b = 103$ ; yields $10607$ ; $10607$ is prime, $10609$ is composite

$a = 107$ and $b = 109$ ; yields $11879$ ; $11879$ is composite, $11881$ is composite

Conclusion: $c$ is either part of a twin-composite pair or a prime-composite pair.

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    $\begingroup$ Can you see a connection between the composite in your prime-composite pairs and one of the twin primes used to generate $c$? $\endgroup$ Dec 18 '20 at 21:54
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    $\begingroup$ If $b$ is the larger of the twin primes, we always have $b^2 = c+2$, whether $c$ is prime or composite. $\endgroup$ Dec 18 '20 at 22:06
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    $\begingroup$ If you want to get the most of out experimental mathematics, I strongly recommend that you learn a programming language. (P.S. and all mathematics is experimental while it is being discovered.) $\endgroup$
    – Rob Arthan
    Dec 18 '20 at 22:33
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    $\begingroup$ Your conjecture holds true up to $a≤10^6$ it also seems $c$ and $c+6n−2$ both also never been an prime for all $n∈\mathbb{Z}_{n\ge0}$. I suggest you go through Pari/GP for programing forprime(a=1,100000,forprime(b=a+2,a+2,forprime(c=a+b+a*b,a+b+a*b,forprime(d=c+2,c+2,print([a,b,c,d]))))) :) $\endgroup$
    – Pruthviraj
    Dec 20 '20 at 12:02
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    $\begingroup$ Check generalizing this problem on MO post, mathoverflow.net/q/379430/149083 $\endgroup$
    – Pruthviraj
    Dec 21 '20 at 17:11
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As @DanielFischer pointed out, the conjecture is true:
$a=b-2$
$\Rightarrow c=(a+b)+(a*b)$
$=(b-2+b)+((b-2)*b)$
$=b^2-2$.
Thus, $c+2=b^2$ is always composite.

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