Gyarmati, K.; Sárközy, A.; Stewart, C. L. On shifted products which are powers. (English) Zbl 1047.11029 Mathematika 49, No. 1-2, 227-230 (2002). 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)]. Reviewer: Florian Luca (Morelia) Cited in 2 ReviewsCited in 7 Documents MSC: 11D45 Counting solutions of Diophantine equations Keywords:shifted products; perfect powers; Diophantine \(m\)-tuples Citations:Zbl 0986.11016; Zbl 1042.11019 PDFBibTeX XMLCite \textit{K. Gyarmati} et al., Mathematika 49, No. 1--2, 227--230 (2002; Zbl 1047.11029) Full Text: DOI References: [1] DOI: 10.2307/2006498 · Zbl 0389.10015 · doi:10.2307/2006498 [2] DOI: 10.1006/jnth.2000.2627 · Zbl 1010.11019 · doi:10.1006/jnth.2000.2627 [3] Bollobás, Extremal Graph Theory (1978) [4] DOI: 10.1093/qmath/20.1.129 · Zbl 0177.06802 · doi:10.1093/qmath/20.1.129 [5] DOI: 10.4064/aa97-1-3 · Zbl 0986.11016 · doi:10.4064/aa97-1-3 [6] Turán, Mat. Fiz. Lapok 48 pp 436– (1941) [7] Kövári, Colloq. Math 3 pp 50– (1954) [8] DOI: 10.1093/qmath/26.1.275 · Zbl 0309.10008 · doi:10.1093/qmath/26.1.275 [9] DOI: 10.1007/BF02411814 · Zbl 0339.10017 · doi:10.1007/BF02411814 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.