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I am asking for references about the following problem. In particular, it is still open? If not, what is the state of the art result?

Problem 1. Let $\Gamma$ be the multiplicative subgroup of $\mathbb{Q}^*$ generated by the set of "shifted primes" $\{p + 1 : p \text{ is a prime number}\}$. Is it true that $\Gamma = \mathbb{Q}^*$?

An affirmative answer would follow from a proof of the Schinzel and Sierpinski's conjecture that for any positive integer $n$ there exists two primes $p$ and $q$ such that $n = (p+1)/(q+1)$. It has already been asked about progress on Schinzel and Sierpinski's conjecture in this MO question, however Problem 1 could be solved without solving Schinzel and Sierpinski's conjecture, so I do not think this question is a duplicate.

It seems to me that currently the best result is $|\mathbb{Q}^* / \,\Gamma| \leq 3$, due to Elliott [1].

Lastly, I ask about the natural generalization with $\{p+h : p \text{ is a prime number}\} \cap \mathbb{Q}^*$, for an integer $h$.

1 Elliott, P. D. T. A. "The multiplicative group of rationals generated by the shifted primes. II." J. Reine Angew. Math. 519 (2000), 59–71.

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