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
ABCDE
353 315 272 120 30
651 599 430 340 240
2487 2420 1384 710 435
2501 2365 1432 1190 1130
2829 2745 1546 1010 850
3723 3152 2460 2345 2270
3973 3395 3230 1652 350
4267 4094 2650 1060 205
4333 3670 3545 1750 1394
4449 4250 2840 699 700
4949 4907 1880 1660 380
5281 5080 3233 1120 1000
5463 5055 3910 1412 410
5491 5400 2634 1770 955
5543 5400 3043 1680 30
5729 5150 4355 1810 1354
6167 5695 4280 2770 542
6609 5984 5000 885 50
6801 6185 4790 3468 1490
7101 6368 5365 2850 1390
7209 7166 2790 1345 160
7339 6635 5440 3052 800
7703 6995 5620 3196 2230
7101 6368 5365 2850 1390
7339 6635 5440 3052 800
7703 6995 5620 3196 2230
8433 7565 5230 4806 4730
8493 7630 5925 4910 524
8517 7815 6100 3440 1642
8577 8230 5236 2905 1050
8637 8012 5780 3695 3450
8373 7123 5670 5500 4450
9137 8570 6180 3285 816
9243 8618 6435 2870 680
9431 7820 6935 5800 5192
9519 8760 6935 1490 1394
9797 8835 6800 5490 2922
9639 8570 7050 5264 305
9877 8864 6485 5660 4840