Abstract
Every positive integern has a representation\(n = t(p_1 + 1)^{\varepsilon _1 } \ldots (p_k + 1)^{\varepsilon _k } \) with thep i prime, eachε i = ±1, and all the prime divisors oft less than an explicit absolute bound. Furthermore, if such a representation is always possible witht=1, then it is also possible with an absolutely bounded number of factorsk.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Davenport, H.: Multiplicative Number Theory. 2nd Ed. Berlin-Heidelberg-New York: Springer. 1980.
Elliott, P. D. T. A.: On two conjectures of Kátai. Acta Arithm.30, 35–59 (1976).
Kátai, I.: On sets characterizing number-theoretical functions (II). The set of “prime plus one” 's is a set of quasi-uniqueness. Acta Arithm.16, 1–4 (1968).
Meyer, J.: Représentation multiplicative des entiers à l'aide de l'ensemblep+1 (II). Astérisque94, 133–142 (1982).
Montgomery, H. L.: Topics in Multiplicative Number Theory. Lect. Notes Math. 227. Berlin-Heidelberg-New York: Springer. 1971.
Montgomery, H. L., Vaughan, R. C.: On the large sieve. Mathematika20, 119–134 (1973).
Page, A.: On the number of primes in an arithmetic progression. Proc. London Math. Soc.39, 116–141 (1935).
Prachar, K.: Primzahlverteilung. Berlin-Göttingen-Heidelberg: Springer. 1957.
Rosser, J. B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math.6, 64–94 (1962).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Elliott, P.D.T.A. On representing integers as products of thep + 1. Monatshefte für Mathematik 97, 85–97 (1984). https://doi.org/10.1007/BF01653238
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01653238