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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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