Abstract
Erdős and Odlyzko proved that odd integers k such that \({k2^n +1}\) is prime for some positive integer n have a positive lower density. We prove that for sufficiently large x, the number of integers k\({\leq}\)x such that k is relatively prime to M and such that \({kM^n+n}\) is prime for some positive integer n is at least C(M)x for some constant C(M) depending only on M.
Similar content being viewed by others
References
Banks, W., Finch, C., Luca, F., Pomerance, C., Stănică, P.: Sierpiński and Carmichael numbers. Trans. Amer. Math. Soc. 367, 355–376 (2015)
Chen, Y.G.: On integers of the form \(k2^{n}+1\). Proc. Amer. Math. Soc. 129, 355–361 (2001)
Chen, Y.G.: On integers of the forms \(k-2^{n}\) and \(k2^{n}+1\). J. Number Theory 89, 121–125 (2001)
Chen, Y.G.: On integers of the forms \(k^{r}-2^{n}\) and \(k^{r}2^{n}+1\). J. Number Theory 98, 310–319 (2003)
Chen, Y.G.: On integers of the forms \(k\pm 2^{n}\) and \(k2^{n}\pm 1\). J. Number Theory 125, 14–25 (2007)
Cilleruelo, J., Luca, F., Pizarro-Madariaga, A.: Carmichael numbers in the sequence \(\{2^{n}k + 1\}_{n>1}\). Math. Comp. 85, 357–377 (2016)
Erdős, P., Odlyzko, A.M.: On the density of odd integers of the form \((p-1)2^{-n}\) and related questions. J. Number Theory 11, 257–263 (1979)
R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer (New York, 2004)
H. Halberstam and H. E. Richert, Sieve Methods, London Math. Soc. Monographs, No. 4, Academic Press (London–New York, 1974)
Luca, F., Stănică, P.: On numbers of the form \(p+2^{n}-n\). J. Comb. Number Theory 6, 157–162 (2016)
W. Sierpiński, Sur un problème concernant les nombres \(k2^{n}+1\), Elem. Math., 15 (1960), 73–74; Corrigendum, 17 (1962), 85.
Sun, X.G., Fang, J.H.: On the density of integers of the form \((p-1)2^{-n}\) in arithmetic progressions. Bull. Austral. Math. Soc. 78, 431–436 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China(11471017) and the National Natural Science Foundation of HuaiHai Institute of Technology (KQ10002).
Rights and permissions
About this article
Cite this article
Sun, XG. Primes of the form \({kM^n+n}\). Acta Math. Hungar. 158, 100–108 (2019). https://doi.org/10.1007/s10474-019-00914-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-019-00914-9