I searched the smallest solution of x4+y4+z4=nw4. Search condition n<100 x>y>z>0 x<5000(about n=2,18,98: x<1000) I made some filters to reduce the number of the search data. 1. Consideration of x4+y4+z4=nw4 mod 16 In the case of n mod 16 =4,5,6,7,8,9,10,11,12,13,14,15,above equation have no solution. 2. Consideration of x4+y4+z4=nw4 mod 25 In the case of n mod 25 =4,5,9,10,14,15,19,20,24,above equation have no solution. Search results 1. Case n=2 There are infinitely many solutions. Using identity x4+y4+(x+y)4=2(x2+xy+y2)2,we can get a parameter solution. One of the solution for x2+xy+y2=1 is (x,y)=(1,0),so we obtain a following parameter solution. (-1+k2)4+(k2+2k)4+(1+2k)4 = 2(1+k+k2)4 k 2 84 + 34 + 54 = 2* 74 3 154 + 84 + 74 = 2* 134 4 84 + 54 + 34 = 2* 74 5 354 + 244 + 114 = 2* 314 6 484 + 354 + 134 = 2* 434 7 214 + 164 + 54 = 2* 194 8 804 + 634 + 174 = 2* 734 9 994 + 804 + 194 = 2* 914 10 404 + 334 + 74 = 2* 374 11 1434 + 1204 + 234 = 2* 1334 12 1684 + 1434 + 254 = 2* 1574 13 654 + 564 + 94 = 2* 614 14 2244 + 1954 + 294 = 2* 2114 15 2554 + 2244 + 314 = 2* 2414 16 964 + 854 + 114 = 2* 914 17 3234 + 2884 + 354 = 2* 3074 18 3604 + 3234 + 374 = 2* 3434 19 1334 + 1204 + 134 = 2* 1274 2. Case n=18 There are infinitely many solutions. In the same way as n=2,we can get a parameter solution. x2+xy+y2=3 is (x,y)=(1,1) (-1+2k+2k2)4+(-2-2k+k2)4+(1+4k+k2)4 = 18(1+k+k2)4 3. Case n=98 There are infinitely many solutions. In the same way as n=2,we can get a parameter solution. x2+xy+y2=7 is (x,y)=(2,1) (-3-2k+2k2)4+(-1+4k+3k2)4+(2+6k+k2)4 = 98(1+k+k2)4 (x,y,z,w)=(-,-,-,-): locally not solvable (x,y,z,w)=(?,?,?,?): solution not found (x,y,z,w)=(*,*,*,*): has many solutions(proved by Elkies) x4+y4+z4=nw4