A cubic triangular number is a positive integer that is simultaneously cubic and triangular. Such a number must therefore satisfy for some positive integers and , where is a triangular number, so
(1)
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But then
(2)
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or
(3)
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According to Catalan's conjecture (now a theorem), the only pair of perfect powers differing by 1 is and , so the unique cubic triangular number has and , implying .