Given that the fourth power of - M + the second power of 4m + the nth power of 2, the second power of M + the nth power of 2 + 5 = 0, and m and N are positive integers, find the value of M and N? There are points. Come and get them!

Given that the fourth power of - M + the second power of 4m + the nth power of 2, the second power of M + the nth power of 2 + 5 = 0, and m and N are positive integers, find the value of M and N? There are points. Come and get them!

-M^4+4M^2+2^N*M^2+2^N+5=0
M^4-4M^2-2^N*M^2-2^N-5=0
(M^4-4M^2+4)-(2^N*M^2+2^N)-9=0
(M^2-2)^2-3^2-2^N(M^2+1)=0
(M^2-2+3)(M^2-2-3)-2^N(M^2+1)=0
(M^2+1)(M^2-5)-2^N(M^2+1)=0
(M^2+1)(M^2-5-2^N)=0
M^2+1>0
So m ^ 2-5-2 ^ n = 0
M^2-2^N=5
Classified discussion
M^2=2^N+5
2 ^ n must be even, so 2 ^ n + 5 must be odd, m ^ 2 must be odd, and M must be odd
1, 2, 4, 8, 16, 32, 64, 128, 256
2 plus 5, 6, 8, 9, 13, 21, 37, 69, 133, 261
Only the third case holds, so n = 2, M = 3