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Surely, but $\,n!+1\,$ and $\,n!+1+n\,$ are not always both primes, as well as $\,lcm(1,\,2,\,\ldots,\,n)+1\,$ and $\,lcm(1,\,2,\,\ldots,\,n)+1+n\,$. So, my point of view is different.