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On shifted products which are powers. (English) Zbl 1047.11029

Let \(V\) denote the set of all pure powers, that is the set of positive integers of the form \(x^k\) with integers \(x\geq 1\) and \(k\geq 2\). Let \(A\) be a subset of \(\{1,2,\dots,N\}\) with the property that \(aa'+1\) is in \(V\) whenever \(a\) and \(a'\) are distinct elements of \(A\). The main result of this paper is the estimate \(\#A<340 (\log N)^2/\log\log N\) on the size of \(A\). The proof is very short, and proceeds as follows. Let \(G\) be the complete graph whose vertices are the points of \(A\). Assign to the edge \(aa'\) of \(G\) the color \(p\), where \(p\leq (\log N)/\log 2\) is a prime, if \(aa'+1\) is a \(p\)th power. If \(A\) has many elements, there exists a prime \(p\) such that \(G\) has many edges colored by the color \(p\). A result of Turán shows that \(G\) contains a complete monochromatic subgraph of size \(r\) with \(r\) “sufficiently large”. On the other hand, results of Dujella on the size of a Diophantine \(m\)-tuple together with a gap principle due to K. Gyarmati [see Acta Arith. 97, 53–65 (2001; Zbl 0986.11016)] show that \(r\) cannot be large, which completes the proof of the Theorem.
Reviewer’s remark: It seems that the result can be improved to \(\#A\ll (\log N/\log\log N)^2\) if instead of the above gap principle one uses the main result of [Y. Bugeaud and A. Dujella, Math. Proc. Camb. Philos. Soc. 135, 1–10 (2003; Zbl 1042.11019)].

MSC:

11D45 Counting solutions of Diophantine equations
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References:

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