Abstract
We derived an asymptotic formula for the number of pairs of integers which are mutually squares. Earlier results dealt with pairs of integers subject to the restriction that they are both odd, co-prime and squrefree. Here we remove all these restrictions and prove (similar to the best known one with restrictions) that the number of such pair of integers upto a large real X is asymptotic to \(\frac{{c{X^2}}}{{\log X}}\) with an absolute constant c which we give explicitly. Our error term is also compatible to the best known one.
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Acknowledgements
This work was supported by Progetti di Ricerca di Interesse Nazionale 2008 “Approssimazione diofantea e teoria algebrica dei numeri”, and Ministerio de Economía y Competitividad of Spain (Grant No. MTM2015-63829-P). Part of this project was realized during the visit of the second author at the Dipartimento di Matematica of the Università Roma Tre financed by Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of Istituto Nazionale di Alta Matematica and part was realized during the visit of the last two authors at the Harish-Chandra Research Institute.
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Chakraborty, K., Jiménez Urroz, J. & Pappalardi, F. Pairs of integers which are mutually squares. Sci. China Math. 60, 1633–1646 (2017). https://doi.org/10.1007/s11425-016-0343-1
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DOI: https://doi.org/10.1007/s11425-016-0343-1