Abstract
We show that the lower density of integers representable as a sum of a prime and a power of two is at least 0.107. We also prove that the set of integers with exactly one representation of the form \(p+2^{k}\) has positive density. Previous results of this kind needed “at most 15” in place of “exactly one”. To achieve these results we introduce a new method. In particular we make use of uneven distribution of sums of a power of two and a reduced residue class.
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Notes
The Cunningham project http://homes.cerias.purdue.edu/~ssw/cun/, see also http://www.mersennewiki.org/index.php/2_Minus_Tables informs about the state of art of factorization of integers of the form \(2^{n}-1\). At the time of writing, for example, the prime factors of \(2^{929}-1\) are not yet known.
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The first named author acknowledges partial support by the Austrian Science Fund (FWF): W1230).
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Elsholtz, C., Schlage-Puchta, JC. On Romanov’s constant. Math. Z. 288, 713–724 (2018). https://doi.org/10.1007/s00209-017-1908-x
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DOI: https://doi.org/10.1007/s00209-017-1908-x