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Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) 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 $\frac{p}{q}$($p,q$ \frac{p}{q}$($p,q$primes) which lies inside$(a,b)$. Does, this problem have any connections with the actual conjecture? I had actually posed this question on MATH.SE (Link : http://math.stackexchange.com/questions/18352/a-conjecture-of-schinzel-and-sierpinski ). I did get a decent answer from Andreis Caicedo, but i would like to have more opinions of Mathematicians from this community. 6 added 4 characters in body Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following: • A conjecture of Schinzel and Sierpinski asserts that every positive rational number$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 had actually posed this question on MATH.SE (Link : http://math.stackexchange.com/questions/18352/a-conjecture-of-schinzel-and-sierpinski ). I did get a decent answer from Andreis Caicedo, but i would like to have more opinions of Mathematicians from this community. 5 deleted 6 characters in body Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) 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$3$. Well, i 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 had actually posed this question on MATH.SE (Link : http://math.stackexchange.com/questions/18352/a-conjecture-of-schinzel-and-sierpinski ). I did get a decent answer from Andreis Caicedo, but i would like to have more opinions of Mathematicians from this community.

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