Composite values of shifted exponentials (Q6175956)

Even though the questions on infinitude of primes of the form \(2^n+1\) or of the form \(2^n-1\) are open, it is clear that the exponents \(n\) for which primaility holds, have natural density \(0\). In passing, we mention that it is widely believed that the number of primes of the form \(2^n-1\) is expected to be infinite, while the number of primes of the form \(2^n+1\) is expected to be only finite. However, it is not even known if there are infinitely many composite values of \(2^p-1\) where \(p\) runs over primes. \textit{C. Hooley} had proved almost five decades back (in his book [Applications of sieve methods to the theory of numbers. Cambridge: Cambridge University Press (1976; Zbl 0327.10044)]) that, for any \(n\), that \(2^n-b\) is composite for almost all \(n\) (in the sense of natural density) under the assumption of GRH and certain other hypotheses. He had also proposed the problem of proving compositeness of \(2^n+5\) for almost all \(n\). Four decades later, Bourgain-Gamburd-Sarnak studied the compositeness of the Markoff numbers using affine sieve methods and the so-called superstrong approximation [\textit{J. Bourgain} et al., C. R., Math., Acad. Sci. Paris 354, No. 2, 131--135 (2016; Zbl 1378.11043)]. Sarnak also connected the affine sieve method to Hooley's question. In the paper under review, the authors prove Hooley's conjecture of compositeness of \(2^n+5\) for almost all \(n\), under the GRH, and another hypothesis that is a type of Brun-Titchmarsh estimate on the average. More generally, they treat the sequence \(a^n-b\) for any integers \(a>1\) and \(b\). Further, they show that even something like \(k\)-almost primeness for any \(k\) can hold only for exponents of density \(0\). More precisely, they prove that for coprime integers \(a>1, b \neq 0\), there is a constant \(c=c(a,b)>0\) such that the natural density of integers \(n\) with \(\omega(a^n-b) \geq c \log \log n\), equals \(1\). Interestingly, the authors' methods allow them to restrict to special exponents like primes. In particular, they are able to prove under the GRH and the average Brun-Titchmarsh estimate alluded to above that for integers \(a>1\) and \(b\) -- other than \((2,1)\) -- the relative density of primes \(p\) such that \(a^p-b\) is composite, equals \(1\). The authors use GRH in a particular situation, and the Brun-Titchmarsh hypothesis they require is a consequence of the pair correlation conjecture. This paper represents a powerful piece of work.