Romanoff’s constant constant
Description of constant
$C_{45}$ is the asymptotic density (if it exists) of the set of odd integers that can be expressed as the sum of a prime number and a power of two.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1/2=0.5$ | Trivial | |
| $<0.5$ | [E1950] | Used covering systems |
| $0.490941$ | [HR2006] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| 0 | Trivial | |
| >0 | [R1934] | |
| $0.0868$ | [CS2004] | |
| $0.0936$ | [P2006] | |
| $0.093627$ | [HS2010] | |
| $0.107648$ | [CE2018] |
Additional comments and links
- In the (unlikely) event that this set fails to have a density, the upper and lower bounds should be interpreted as bounds on the upper and lower densities, respectively.
- Variants of this set are studied in Erdős problems #205, #244 and #851.
References
- [CS2004] Y.G. Chen and X.G. Sun, “On Romanoff’s constant,” J. Number Theory, 106 (2004), 275–284.
- [CE2018] C. Elsholtz and J.-C. Schlage-Puchta, “On the density of sums of primes and powers of two,” Acta Arithmetica, 183 (2018), 1-20.
- [E1950] P. Erdős, “On integers of the form $2^k+p$ and some related problems,” Summa Brasil. Math., 2 (1950), 113–123.
- [HR2006] L. Habsieger and X. Roblot, “On integers of the form $p + 2^k$,” Acta Arithmetica, 122 (2006), 45–50.
- [HS2010] L. Habsieger and R. Sivak-Fischler, “A new lower bound for Romanoff’s constant,” Journal of Number Theory, 130 (2010).
- [P2006] J. Pintz, “A note on Romanoff’s constant,” Acta Mathematica Hungarica, 112 (2006), 1-14.
- [R1934] N. P. Romanoff, “Über einige Sätze der additiven Zahlentheorie,” Math. Ann., 109 (1934), 668–678.
Contribution notes
Prepared with assistance from Gemini.