Imperfection... again

Number Theory Level pending

Are there any two positive integers a a ; b b such that the following expression is a perfect cube?

( 5 a + b − 6 ) 3 + ( 5 b − a + 6 ) 3 + 48 ( 2 a − b + 5 ) ( a + b ) ( 2 b + a − 5 ) \large \displaystyle (5a + b - 6)^3 + (5b - a + 6)^3 + 48(2a - b + 5)(a + b)(2b + a - 5)

Bonus

  • If there are solutions, finitely or infinitely, find all of them.

  • If there aren't any solutions, prove why.

Yes, there are finite solutions. Yes, there are infinite solutions. No, there aren't any solutions.

Thành Đạt Lê by Thành Đạt Lê , 17, Singapore

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