On real quadratic fields with odd class number (Q1298140)

For some fundamental discriminant \(\Delta\), let \(h(\Delta)\) denote the class number of the quadratic field \(\mathbb{Q}(\sqrt \Delta)\). The authors show that there is an infinity of odd positive fundamental discriminants \(\Delta\) such that \(\Delta+4\) is a fundamental discriminant, too, and both \(h (\Delta)\) and \(h(\Delta+4)\) are odd. More precisely, the number of \(\Delta\leq x\) with this property exceeds \(cx/(\log x)^2\) as \(x\to\infty\), where \(c\) is some suitable absolute positive constant. By genus theory, the problem is reduced to the question of finding an infinity of positive integers \(n\) such that both \(n\) and \(n+4\) are either prime or the product of two primes congruent to 3 mod 4. The main tools of the proof are a weighted inequality of the type used by \textit{J.-R. Chen} and \textit{J. Wu} [Sci. Sin. 16, 157-176 (1973; Zbl 0319.10056)] and \textit{J. Wu} [Acta Arith. 55, 365-394 (1990; Zbl 0662.10033)], the linear sieve of Rosser-Iwaniec involving well-factorable functions in the sieve remainder term [Acta Arith. 37, 307-320 (1980; Zbl 0444.10038)] and the results of \textit{E. Bombieri}, \textit{J. B. Friedlander} and \textit{H. Iwaniec} on the equidistribution of primes in residue classes to large moduli [Acta. Math. 156, 203-251 (1986; Zbl 0588.10042)].