On the number of distinct prime factors of $$nj+a^hk$$.(English)Zbl 1311.11097

T. Tao [J. Aust. Math. Soc. 91, No. 3, 405–413 (2011; Zbl 1251.11089)] proved that, for any integer $$K \geq 2$$, a positive proportion of the primes in the interval $$[x, (1+K^{-1}) x]$$ have the property that $$|pj \pm a^h k|$$ is composite for every $$2 \leq a \leq K$$, $$1 \leq j, k \leq K$$ and $$1 \leq h \leq K \log x$$. This result has the nice consequence that it is not possible to tell whether an integer is a prime without reading all of its base $$a$$ digits.
The present paper generalises Tao’s result in certain aspects showing that there exists at least $$x^{1-\varepsilon}$$ integers $$n \in [x, (1+K^{-1}) x]$$ which have the property that $$\omega(n j \pm a^h k) \geq (\log \log \log x)^{1/3-\varepsilon}$$ for every $$2 \leq a \leq K$$, $$1 \leq j, k \leq K$$ and $$1 \leq h \leq K \log x$$, where $$\omega(m)$$ denotes the number of distinct prime factors of $$m$$, i.e., the author shows that there are quite many numbers $$n$$ (though fewer than in Tao’s result) such that $$n j \pm a^h k$$ are not just barely composite but all of them have relatively many prime factors.
The proof is a refined and more efficient version of Tao’s method which uses incomplete covers of $$\mathbb Z$$.

MSC:

 11P32 Goldbach-type theorems; other additive questions involving primes 11A07 Congruences; primitive roots; residue systems 11B25 Arithmetic progressions 11N36 Applications of sieve methods

Zbl 1251.11089
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