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On shifted products which are powers

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

K. Gyarmati ,
A. Sárközy  and
C. L. Stewart
K. Gyarmati
Affiliation:
Department of Algebra and Number Theory, University Eötvos Loránd, H-1053 Budapest, Hungary, E-mail: gykati@cs.elte.hu
A. Sárközy
Affiliation:
Department of Algebra and Number Theory, University Eötvos Loránd, H-1053 Budapest, Hungary. E-mail: sarkozy@cs.elte.hu
C. L. Stewart
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3GI. E-mail: cstewart@watservl.uwaterloo.ca
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Extract

Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj+1 is a square for 1≤i<j≤4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grinstead [4] gave alternative proofs. The construction of such sets originated with Diophantus who studied the problem when the ai are rational numbers. It is conjectured that there do not exist five positive integers whose pairwise products are all one less than the square of an integer. Recently Dujella [3] proved that there do not exist nine such integers. In this note we address the following related problem. Let V denote the set of pure powers, that is, the set of positive integers of the form xk with x and k positive integers and k>1. How large can a set of positive integers A be if aa′ + 1 is in V whenever a and a′ are distinct integers from A? We expect that there is an absolute bound for |A|, the cardinality of A. While we have not been able to establish this result, we have been able to prove that such sets cannot be very dense.

MSC classification

Secondary: 11D45: Counting solutions of Diophantine equations
Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 227 - 230
Copyright
Copyright © University College London 2002

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References

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