## On the density of odd integers of the form $$(p-1)2^{-n}$$ and related questions.(English)Zbl 0405.10036

Given $$k$$ primes $$p_1,\dots,p_k$$, write $$p-1=p_1^{a_1}\dots p_k^{a_k}s_p$$, where $$s_p$$ is coprime to $$P=p_1p_2\dots p_k$$. It is proved that the sequence of numbers occurring as $$s_p$$ for some prime $$p$$ has positive lower density. The most interesting unsolved problem is whether this sequence ($$s_p$$) can contain all numbers, coprime to $$P$$; concerning this question some numerical data are given.

### MSC:

 11B05 Density, gaps, topology 11N05 Distribution of primes 11N35 Sieves

### Keywords:

primes of special form; divisors; sieve methods; density
Full Text:

### References:

 [1] Bombieri, E., Le grand crible dans la théorie analytique des nombres, Astérisque, 18 (1974) · Zbl 0292.10035 [2] Davenport, H., (Multiplicative Number Theory (1967), Markham: Markham Chicago) · Zbl 0159.06303 [3] Erdös, P., On integers of the form $$2^k + p$$ and some related problems, Summa Brasil Math., 2, 113-123 (1950) · Zbl 0041.36808 [4] Halberstam, H.; Richert, H.-E, (Sieve Methods (1974), Academic Press: Academic Press London/New York) · Zbl 0298.10026 [5] Prachar, K., Primzahlverteilung (1957), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0080.25901 [6] Robinson, R. M., A report on primes of the form $$k$$ · $$2^n + 1$$ and on factors of Fermat numbers, (Proc. Amer. Math. Soc., 9 (1958)), 673-681 · Zbl 0092.27505 [7] Selfridge, J. L., Solution to problem 4995, Amer. Math. Monthly, 70, 101 (1963) [8] J. L. Selfridge; J. L. Selfridge [9] Sierpiński, W., Sur un problème concernant les nombres $$k$$ · $$2^n + 1$$, Elem. Math., 15, 73-74 (1960), Corrigendum 17 (1962), 85 · Zbl 0093.04602 [10] Sierpiński, W., (Elementary Theory of Numbers (1964), Polish Scientific Publishers: Polish Scientific Publishers Warsaw) · Zbl 0638.10001 [11] Sierpiński, W., (250 Problems in Elementary Number Theory (1970), Elsevier: Elsevier New York) · Zbl 0211.37201
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