I searched the  some primitive solutions of A4 = B4 +C4 +D4 +E4.

Search condition
    A<10000
    B>=C>=D>=E
    gcd(A,B,C,D,E)=1

I made some filters to reduce the number of the search data.


    1. Consideration of A4 = B4 +C4 +D4 +E4 mod 16

       Generally,N4= 0 or 1 mod 16.
       If A is even number, all of (B,C,D,E) become the even number.
       But, this is not the primitive solution.

       So,If we search the primitive solutions,A must be odd number.
       Only one of (B,C,D,E) is odd number and other rest of (B,C,D,E) must be even number.
       
    2. Consideration of A4 = B4 +C4 +D4 +E4 mod 5

       Generally,N4= 0 or 1 mod 5.
       If A is divisible by 5, all of (B,C,D,E) are divisible by 5.
       But, this is not the primitive solution.

       So,If we search the primitive solutions,A must not be divisible by 5.
       Only one of (B,C,D,E) is not divisible by 5 and other rest of (B,C,D,E) must be divisible by 5.






                       
                                      

                                       A4 = B4 +C4 +D4 +E4