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Melvyn Nathanson, in his book Methods in Number Theory (Chapter 8: Prime Number's) states the following:

A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can be represented as a quotient of shifted primes, that $x=\frac{p+1}{q+1}$ for primes $p$ and $q$. It is known that the set of shifted primes, generates a subgroup of the multiplicative group of rational numbers of index at most $3$.

I would like to know what progress has been made regarding this problem and why is this conjecture important. Since it generates a subgroup, does the subgroup which it generates have any special properties?

I had actually posed a problem which asks us to prove that given any interval $(a,b)$ there is a rational of the form $\frac{p}{q}$ ($p,q$ primes) which lies inside $(a,b)$. Does, this problem have any connections with the actual conjecture?

I actually posted this question on MO. Interested users can see this link:

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    *Any* collection of nonzero rational numbers generates "a subgroup of the multiplicative group of rational numbers." In particular, "this generates a subgroup of the multiplicative group of rational numbers" requires no proof.2011-01-20
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    @Arturo: Oh, yes.2011-01-20

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