Primes, products and polynomials (Q1401435)

This paper contains important progress on solutions of Diophantine equations with few prime variables. Let \(f\in \mathbb{Z}[x]\) be an irreducible polynomial with positive leading coefficient. It was conjectured by Polignac, Hardy and Littlewood, Schinzel, Bateman and Horn that the values \(f(n), n \in \mathbb{N}\), are prime infinitely often. This has only been proved for linear polynomials. An even more difficult problem is to ask for prime values of \(f(q)\), where \(q\) is itself a prime. While it is not known whether \(p=2q^2+5\) (i.e. \(2= \frac{p+1} {q^2+3}\)) has infinitely many prime solutions, the author shows that \[ 2= \frac{p_1+1}{q_1^2+3} \frac{p_2+1} {q_2^2+3} \frac{q_3^2+3}{p_3+1} \] has infinitely many solutions with prime \(p_i, q_i\). Similarly for \(2= \frac{p_1+1}{q_1^3+5} \frac{p_2+1}{q_2^3+5} \frac{q_3^3+5}{p_3+1}\). Another important result is Theorem 3: Let \(f\) be a non-constant polynomial with integer coefficients. Let \(c>0\). Then \((x_1+1)f(x_2)-(x_3+1)f(x_4)=0\) has infinitely many solutions with \(x_2 > x_4^c\) and all \(x_i\) are prime. Note that the number of variables is 4, i.e. constant, while the degree can be arbitrarily large. Related results on the Goldbach-Waring problem require that the number of variables increases with the degree. Also note that the twin prime problem \(x_1-x_2+2=0\) is closely related. All these results are deduced from Theorem 4, which states that the integers that can be represented as \(\frac{p+1}{f(q)}\) have density at least \(1/4\). The proofs use methods from analytic number theory such as Selberg's sieve and Linnik's dispersion method.