Prime Sequence Conjecture

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.

• Can you see a connection between the composite in your prime-composite pairs and one of the twin primes used to generate $c$? Dec 18 '20 at 21:54
• If $b$ is the larger of the twin primes, we always have $b^2 = c+2$, whether $c$ is prime or composite. Dec 18 '20 at 22:06
• 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.) Dec 18 '20 at 22:33
• 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]))))) :) Dec 20 '20 at 12:02
• Check generalizing this problem on MO post, mathoverflow.net/q/379430/149083 Dec 21 '20 at 17:11

$$a=b-2$$
$$\Rightarrow c=(a+b)+(a*b)$$
$$=(b-2+b)+((b-2)*b)$$
$$=b^2-2$$.
Thus, $$c+2=b^2$$ is always composite.