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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

Citations:

Zbl 1251.11089
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References:

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