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Perfect powers in arithmetical progression. II. (English) Zbl 0763.11014
This is a continuation of work reported in the authors’ joint papers [Perfect powers in arithmetic progression, J. Madras Univ., Sect. B 51, 173-180 (1988) and Compos. Math. 75, 307-344 (1990; Zbl 0708.11021)]. Suppose \(\ell\geq 3\) is a prime, \(m\) and \(d\) are positive co-prime integers with \(k\geq 3\) and such that each of the numbers \(m,m+d,\dots,m+(k-1)d\) is a perfect \(\ell\)-th power. Further, let \(d_ 1\) denote the largest divisor of \(d\) with the property that all its prime factors are \(\equiv 1\pmod \ell\) and let \(m_ 1\) be similarly defined. The main results of the present paper are:
Theorem 1. Let \(\varepsilon>0\). There exists an effectively computable (e.c.) number \(C_ 2\) depending on \(\varepsilon\) only, such that \(k\geq C_ 2\) implies that \(2^{\omega(d_ 1)}>(1-\varepsilon)k\).
Theorem 2. a) There exists an e.c. absolute constant \(C_ 5>0\) such that \(\log(m+1)\geq C_ 5 k^ 2\).
b) There exist e.c. absolute constants \(C_ 6\) and \(C_ 7>0\) such that \(k\geq C_ 6\) implies that \[ \ell^{\omega(m_ 1)}\geq C_ 7 k,\;p(m_ 1)>k\ell,\quad\text{and}\quad Q(m_ 1)\geq C_ 7 k^{1+\omega(m_ 1)}. \] Theorem 3. There exists an e.c. absolute constant \(C_ 8\) such that \(k\geq C_ 8\) implies \(P(m-d)\geq k\).
Here \(\omega(n)\), \(p(n)\), \(P(n)\), and \(Q(n)\) are respectively the number of distinct prime factors, the least prime factor, the greatest prime factor, and the greatest square free factor of \(n\).

MSC:
11D61 Exponential Diophantine equations
11B25 Arithmetic progressions
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