We prove that there are only finitely many positive integers $m$ such that there is some integer $t$ such that $|n^2+n-m|$ is 1 or a prime for all $n\in [t+1, t+\sqrt{m}]$, thus solving a problem of Byeon and Stark.
@article{107908,
author = {Jan-Christoph Schlage-Puchta},
title = {Finiteness of a class of Rabinowitsch polynomials},
journal = {Archivum Mathematicum},
volume = {040},
year = {2004},
pages = {259-261},
zbl = {1122.11070},
mrnumber = {2107020},
language = {en},
url = {http://dml.mathdoc.fr/item/107908}
}
Schlage-Puchta, Jan-Christoph. Finiteness of a class of Rabinowitsch polynomials. Archivum Mathematicum, Tome 040 (2004) pp. 259-261. http://gdmltest.u-ga.fr/item/107908/
On the Finiteness of Certain Rabinowitsch Polynomials, J. Number Theory 94 (2002), 177–180. | Article | MR 1904967 | Zbl 1033.11010
On the Finiteness of Certain Rabinowitsch Polynomials. II, J. Number Theory 99 (2003), 219–221. | Article | MR 1957253 | Zbl 1033.11010
Zero-free regions for Dirichlet $L$-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), 265–338. (1992) | MR 1143227 | Zbl 0739.11033
Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern, J. Reine Angew. Mathematik 142 (1913), 153–164. (1913)