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if sum \sqrt[3]{a}+\sqrt[3]{b} is rational then a,b are perfect cubes

Source: 1992 ITAMO p6

January 31, 2020

perfect cuberationalnumber theory

Problem Statement

Let

a

and

b

be integers. Prove that if

\sqrt[3]{a}+\sqrt[3]{b}

is a rational number, then both

a

and

b

are perfect cubes.