Hello

n=5p (where p is an integer) is a necessary condition but I do not know if it is a sufficient condition

If n=5p+1

Then n^8=(5p+1)^8=5P+1

Therefore 4n^8+1=20P+5 is divisible by 5 => not prime

If n=5p+2

Then n^8=(5p+2)^8=5P+2^8=5P+256

Therefore 4n^8+1=20P+1025 is divisible by 5 => not prime

If n=5p+3

Then n^8=(5p+3)^8=5P+3^8=5P+6561

Therefore 4n^8+1=20P+26245 is divisible by 5 => not prime

If n=5p+4

Then n^8=(5p+4)^8=5P+4^8=5P+65536

Therefore 4n^8+1=20P+262145 is divisible by 5 => not prime