If d is a positive integer, then the 9 d 4 + 132 d 2...

Lovellss

Lovellss

Answered

2022-06-25

If d is a positive integer, then the 9 d 4 + 132 d 2 16 is an integer or an irrational number? This question appears in my attempt to prove that the only triangle with integers for sides and area in arithmetic progression has sides ( 3 , 4 , 5 ) and area 6.

Answer & Explanation

Schetterai

Schetterai

Expert

2022-06-26Added 25 answers

9 d 4 + 132 d 2 16 = ( 3 d 2 + 22 ) 2 500. As the next biggest square below n 2 that is of the same parity as n 2 is ( n 2 ) 2 = n 2 4 ( n 4 ), we conclude that 9 d 4 + 132 d 2 16 can at most be a square when 4 ( 3 d 2 + 22 1 ) 500, i.e., d < 6. This leaves finitely many cases for manual checking.

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