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I got this statement by upcoming mathematician Prof. Gandhi from BITS:

"All twin primes from $17$ who are the smallest elements of a pair of twin primes, can be rewritten as: $(a + b + 1)$ such that $a$, $(a+2)$, $b$ and $(b+2)$ are primes".

I checked with my own examples, $17 = 5+11+1$; $29 = 11+17+1$; 41=11+29+1; $59 = 29+29+1=17+41+1$, $71=11+59+1$

So far I have not found any counterexamples.

Could you discuss or comment on cited conjecture? I am looking for discussion.

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closed as not a real question by Raskolnikov, Jyrki Lahtonen, Kannappan Sampath, Asaf Karagila, Henning Makholm Feb 27 '12 at 0:59

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This website is specifically designed to preclude discussion. But anyway the conjecture is almost surely true and absolutely surely hopeless. I say it's almost surely true because we have good conjectural estimates on the number of twin primes up to any given $N$, and then the same heuristics that make the Goldbach conjecture a dead certainty apply. –  Gerry Myerson Feb 27 '12 at 0:01