Find the largest positive integer n < 30 such that 1 2 (n^8+3 n^4-4 ) is not divisible by the square of any prime number.

Step-by-step solution

Let f(n)= 1 2 (n^8+3 n^4-4 )= 1 2 (n-1)(n+1) (n^2+1 ) ( (n^2+1 )+1 ) ((n-1)^2+1 ) For n=2 k+1 f(2 k+1)= 1 2 (2 k) 2(k+1) (4 k^2+4 k+2 ) Clearly 4 f(2 k+1) Only even cases n=28, f(28)= 1 2 27 29 (28^2+1 ) (27^2+1 ) (29^2+1 ) Clearly 3^2 f(28)n=26, f(26)= 1 2 25 27 (25^2+1 ) (26^2+1 ) (27^2+1 ) Clearly 5^2 f(26)n=24, 1 2 23 25 (24^2+1 ) (25^2+1 ) (26^2+1 ) Again 5^2 f(26) aligned & n=22, f(22)= 1 2 21 23 (22^2+1 ) (21^2+1 ) (23^2+1 ) & 5 |22^2+1,5 | 23^2+1 & 52 f(22) & n=20, f(20)= 1 2 (19 21) (20^2+1 ) (21^2+1 ) (19