Find the largest positive integer $n$ < 30 such that ( 1 8 ) $n$ + 3n 4 − 4 is not divisible by the square of any prime number.

Step 1: Let f ( $n$ ) =

Step 2: n + 3n 4 − 4 )

Step 3: ( $n$ − 1)( $n$ + 1) $n$ 2 + 1 ) ( ( $n$ + 1) + 1) ( ( $n$ − 1) + 1)

Step 4: 2 2

Step 5: For $n$ = 2k + 1

Step 6: => f ( 2k + 1) =

Step 7: ( 2k ) 2 ( $k$ + 1) 4k 2 + 4k + 2 \dots )

Step 8: Clearly 4 | f(2k + 1)

Step 9: => Only even cases

Step 10: => $n$ = 28, f ( 28 ) =

Step 11: ( )(

Step 12: \times 27 \times 29 \times 282 + 1 272 + 1 292 + 1 )( )

Step 13: Clearly 32 | f(28)

Step 14: n = 26, f ( 26 ) =

Step 15: ( )(

Step 16: \times 25 \times 27 \times 252 + 1 262 + 1 272 + 1 )( )

Step 17: Clearly 52 | f(26)

Step 18: ·23 \times 25· 242 + 1 252 + 1 262 + 1 )( )

Step 19: Again 52 | f(26)

Step 20: n = 22, f ( 22 ) =

Step 21: ( )(

Step 22: \times 21\times 23 \times 222 + 1 212 + 1 232 + 1 )( )

Step 23: 5 | 222 + 1, 5 | 232 + 1

Step 24: => 52 | f(22)

Step 25: n = 20, f ( 20 ) =

Step 26: ( )(

Step 27: (19 \times 21) 202 + 1 212 + 1 192 + 1 )( )

Step 28: Not divisible by any prime square

Step 29: => $n$ = 20