Abstract
Diophantine sets, i.e. sets of positive integers A with the property that the product of any two distinct elements of A increased by 1 is a perfect square, have a vast literature, dating back to Diophantus of Alexandria. The most important result states that such sets A can have at most five elements, and there are only finitely many of them with five elements. Beside this, there are a large number of finiteness results, concerning the original problem and some of its many variants. In this paper we introduce the notion of, and prove finiteness results on so called (F, m)-Diophantine sets A, where F is a bivariate polynomial with integer coefficients, and instead of requiring \(ab+1\) to be a square for all distinct \(a,b\in A\), the numbers F(a, b) should be full m-th powers. The particular choice \(F(x,y)=xy+1\) and \(m=2\) gives back the original problem.
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We are grateful to the Referee for the helpful comments and remarks.
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Communicated by J. Schoißengeier.
The research was supported in part by the University of Debrecen, by the János Bolyai Scholarship of the Hungarian Academy of Sciences (A.B.) and by grants K100339 (A.B., L.H., Sz.T.), NK104208 (A.B., Sz.T.), and NK101680 (L.H.) of the Hungarian National Foundation for Scientific Research. This work was partially supported by the European Union and the European Social Fund through project Supercomputer, the national virtual lab (Grant No.: TAMOP-4.2.2.C-11/1/KONV-2012-0010). The paper was also supported by the TÁMOP-4.2.2.C-11/1/KONV-2012-0001 project. The project has been supported by the European Union, co-financed by the European Social Fund. A.D. has been supported by Croatian Science Foundation under the project no. 6422.
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Bérczes, A., Dujella, A., Hajdu, L. et al. Finiteness results for F-Diophantine sets. Monatsh Math 180, 469–484 (2016). https://doi.org/10.1007/s00605-015-0824-6
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DOI: https://doi.org/10.1007/s00605-015-0824-6