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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 $\frac{p}{q}$ ($p,q$ 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 : https://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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    $\begingroup$ The question of approximating real numbers by ratios of prime numbers is interesting. One can show that for any positive real number $u$, there are infinitely many pairs of primes $(p,q)$ so that $|u - p/q|<Cq^{-0.465}$, where $C$ depends only on $u$. To prove this, you can use Baker and Harman's result; they show that there exists a prime number between $n$ and $n+n^{0.535}$. $\endgroup$
    – Hej
    Commented Jan 29, 2011 at 20:33
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    $\begingroup$ That would surely do it. From the Prime Number theorem it is known that for every $\epsilon>0$ there is an $N$ so that there is always a prime in $(n,(1+\epsilon)n)$ for $n>N$. That is also enough. $\endgroup$
    – Aaron Meyerowitz
    Commented Jan 29, 2011 at 20:44
  • $\begingroup$ @Aaron, you are right. Actually, I am interested in knowing 'how well' we can approximate real numbers by ratios of primes, and so it seems that the result I mentioned is the best we can have without using RH. It is conjectured that for any $u>0$ and any $\epsilon>0$, there exist infinity many pairs of primes $(p,q)$ so that $|u-p/q|<\epsilon \log(q)/q$. This is definitely true for $u=1$ since "small gaps between primes exist" (see the paper by Goldston, et. al). $\endgroup$
    – Hej
    Commented Jan 30, 2011 at 1:37

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The question can be written as follows: Given two positive integers $a$ and $b$, do there exist primes $p$ and $q$ such that $$aq-bp=b-a?$$ You would expect there to be not just one such pair of primes, but infinitely many pairs. For instance, if $a=2$ and $b=1$, then $q$ is a Sophie Germain prime, and everyone expects there to be infinitely many of those. Moreover, you should be able to replace the right side of the equation with a constant $c$, i.e., $$aq-bp=c.$$ The twin prime conjecture says that there are infinitely many solutions when $a=b=1$ and $c=2$. Polignac's conjecture implies infinitely many solutions when $a=b=1$ and for every even value of $c$. In general you should expect infinitely many solutions when there isn't some obvious congruence that forces finiteness; for instance obviously $a=b=c=1$ only has one solution. Moreover, it's natural to expect a specific slowly decreasing density of solutions using a heuristic estimate derived from the prime number theorem.

This question for all suitable $a$, $b$, and $c$ is in turn a special case of yet more general questions about linear patterns in the prime numbers. For instance, the statement that there are infinitely many arithmetic progressions of length 3 in the primes is the statement that there are infinitely many solutions to $$p-q = q - r > 0.$$ Now, it's a famous theorem of Tao and Green that there are infinitely many arithmetic progressions of primes of arbitrary length. Later Tao and Green did a more systematic study that established the existence of all kinds of linear patterns in the prime numbers. However, the Sieprinski-Schnizel conjecture, and its generalization in the previous paragraph, are part of the "rank 1 case" that they did not solve. (These are just my mental notes from a survey talk by Terry Tao that I once attended.) If they could have done the rank 1 case, it would have included the twin prime conjecture and I think that it would have implied the asymptotic Goldbach conjecture too, so that would have been even more amazing than what they did accomplish.

I have no idea whether this remaining rank 1 case is the same class of question as the Tao-Green results, but just harder; or whether it is so much harder that it is in a different class. Let's optimistically say that it's the former. If so, then what makes the Schinzel-Sierpinski conjecture interesting is that you should always expect infinitely many solutions in prime numbers to linear equations, unless there are only finitely many solutions because of a simple congruence. And I might say that the Tao-Green results are the main recent progress, even though they answered different questions.

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