Computations concerning primes and powers of two (Q797630)

Let \(a(n)=\#\{m: m\leq n\), \(m=p+2^k\}\) and \(\delta(N,n)=(a(N+n) - a(N))/n\). It is known [see \textit{N. P. Romanoff}, Math. Ann. 109, 668--678 (1934; Zbl 0009.00801); \textit{J. G. van der Corput}, Simon Stevin 27, 99--105 (1950; Zbl 0037.16901); \textit{P. Erdős}, Summa Brasil. Math. 2, 113--123 (1950; Zbl 0041.36807)] that \[ 0<\lim \inf \delta(0,n)\leq \lim \sup \delta(0,n)<\tfrac12, \] and it is conjectured that \(\delta = \lim \delta(0,n)\) exists. The author computed \(\delta(0,n)\) for \(n\leq 2^{27}\), and \(\delta(N,n)\) for \(n\leq 10^7\), with various \(N\) near \(2^k\) up to \(k=31\). The results are given in graphs which show that \(\delta(0,n)\) has rather prominent local mins. and maxs. corresponding to\(n\) just before and just after a power of 2. These mins. and maxs. near \(2^k\) are given to 5 decimal places for \(9\leq k\leq 26\). It would have been useful to have the exact locations of these mins. and maxs.; similarly it would be of some interest to have various exact values of \(a(n)\). The calculations are extrapolated leading to a suggested value of \(0.434\dots\) for \(\delta\). Computations based on a probabilistic model suggested by Bombieri also lead to a similar value for \(\delta\).