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.

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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.

Schetterai · Accepted Answer

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  &amp;lt;  6. This leaves finitely many cases for manual checking.

If d is a positive integer, then the <msqrt> 9 d 4 </msup>

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