Cipu, MihaiTrudgian, Tim2016-08-242016-08-240065-1036http://hdl.handle.net/1885/107296We consider Diophantine quintuples {a,b,c,d,e}. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most 5.441⋅10²⁶ Diophantine quintuples.© Instytut Matematyczny PAN, 2016. http://www.sherpa.ac.uk/romeo/issn/0065-1036/..."author can archive post-print (ie final draft post-refereeing)" from SHERPA/RoMEO site (as at 25/08/16).Diophantine m-tuplesPell equationslinear forms in logarithms2016-05-1810.4064/aa8254-2-2016