On a question of Mordell

A Diophantine equation is a polynomial equation to which one seeks solutions in integers. There is a notable disparity between the difficulty of stating Diophantine equations and that of solving them. This feature was formalized in the 20th century by Matiyasevich's negative answer to Hilbert's tenth problem: It is impossible to tell whether some Diophantine equations have solutions or not. One need not look very far to find examples whose status is unknown. A striking example was noted by Mordell in 1953: The equation x³+y³+z³=3 has the solutions (1,1,1 ) and (4,4 ,−5 ) (and permutations); are there any others? This paper concludes a 65-y search with an affirmative answer to Mordell's question and strongly supports a related conjecture of Heath-Brown.