A new article  looks at the problem of determining the proportion of odd numbers that can be written as a power of 2 and a prime. A. de Polignac conjectured in 1849 that all odd numbers have this form.
A counterexample is 127, and so apparently the conjecture was that every odd number is either prime or a power of 2 plus a prime . Alternatively, you could say that a prime is a prime plus a power of 2 where the exponent is -∞.
The smallest counterexample to Polignac’s conjecture is 905. It is clearly not prime, nor are any of the values you get by subtracting powers of 2: 393, 649, 777, 841, 873, 889, 897, 901, and 903.
The proportion of numbers of the form 2n plus a prime is known to be between 0.09368 and 0.49095. In  the authors give evidence that the proportion is around 0.437.
You could generalize Polignac’s conjecture by asking how many powers of 2 you need to add to a prime in order to represent any odd number. Clearly every odd number x is the sum of some number of powers of 2 since you can write numbers in binary. But is there a finite upper bound k on the number of powers of 2 needed, independent of x? I imagine someone has already asked this question.
Polignac conjectured that k = 1. The example x = 905 above shows that k = 1 won’t work. Would k = 2 work? For example, 905 = 2 + 16 + 887 and 887 is prime. Apparently k = 2 is sufficient for numbers less than 1,000,000,000.
More posts on number theory
 Gianna M. Del Corso, Ilaria Del Corso, Roberto Dvornicich, and Francesco Romani. On computing the density of integers of the form 2n + p. Mathematics of Computation. https://doi.org/10.1090/mcom/3537 Article electronically published on April 28, 2020