1.Introduction

Sastry(1934) found a parameter solution for 5.1.5 equations.

(75v5-u5)5+(u5+25v5)5+(u5-25v5)5+(10u3v2)5+(50uv4)5=(u5+75v5)5
http://mathworld.wolfram.com/DiophantineEquation5thPowers.html 


I found many parameter solutions for 5.4.6 equations.


2. Result
          
         There are many parameter solutions for 5.4.6 equations. 
         
        (x5+65y5)5+(x5-65y5)5+(x5+90y5)5+(x5-90y5)5
        =(x5+10y5)5+(x5-10y5)5+(x5+85y5)5+(x5-85y5)5+(10x3y2)5+(50xy4)5

        (x5+155y5)5+(x5-155y5)5+(x5+80y5)5+(x5-80y5)5
        =(x5+145y5)5+(x5-145y5)5+(x5+120y5)5+(x5-120y5)5-(10x3y2)5-(50xy4)5

        (x5+220y5)5+(x5-220y5)5+(x5+145y5)5+(x5-145y5)5
        =(x5+205y5)5+(x5-205y5)5+(x5+180y5)5+(x5-180y5)5-(10x3y2)5-(50xy4)5

        (x5+410y5)5+(x5-410y5)5+(x5+335y5)5+(x5-335y5)5
        =(x5+390y5)5+(x5-390y5)5+(x5+365y5)5+(x5-365y5)5-(10x3y2)5-(50xy4)5

        (x5+790y5)5+(x5-790y5)5+(x5+60y5)5+(x5-60y5)5
        =(x5+740y5)5+(x5-740y5)5+(x5+490y5)5+(x5-490y5)5-(20x3y2)5+(200xy4)5

        (x5+900y5)5+(x5-900y5)5+(x5+400y5)5+(x5-400y5)5
        =(x5+800y5)5+(x5-800y5)5+(x5+700y5)5+(x5-700y5)5-(20x3y2)5+(200xy4)5

        (x5+1035y5)5+(x5-1035y5)5+(x5+185y5)5+(x5-185y5)5
        =(x5+1015y5)5+(x5-1015y5)5+(x5+485y5)5+(x5-485y5)5-(20x3y2)5+(200xy4)5

        (x5+1140y5)5+(x5-1140y5)5+(x5+915y5)5+(x5-915y5)5
        =(x5+840y5)5+(x5-840y5)5+(x5+465y5)5+(x5-465y5)5+(30x3y2)5+(450xy4)5

        (x5+2190y5)5+(x5-2190y5)5+(x5+615y5)5+(x5-615y5)5
        =(x5+1965y5)5+(x5-1965y5)5+(x5+1590y5)5+(x5-1590y5)5-(30x3y2)5+(450xy4)5

        (x5+2220y5)5+(x5-2220y5)5+(x5+2020y5)5+(x5-2020y5)5
        =(x5+1580y5)5+(x5-1580y5)5+(x5+1180y5)5+(x5-1180y5)5+(40x3y2)5+(800xy4)5

        (x5+2220y5)5+(x5-2220y5)5+(x5+1810y5)5+(x5-1810y5)5
        =(x5+2060y5)5+(x5-2060y5)5+(x5+2030y5)5+(x5-2030y5)5-(20x3y2)5+(200xy4)5

        (x5+2360y5)5+(x5-2360y5)5+(x5+1560y5)5+(x5-1560y5)5
        =(x5+1240y5)5+(x5-1240y5)5+(x5+1160y5)5+(x5-1160y5)5+(40x3y2)5+(800xy4)5

        (x5+2430y5)5+(x5-2430y5)5+(x5+1630y5)5+(x5-1630y5)5
        =(x5+1730y5)5+(x5-1730y5)5+(x5+670y5)5+(x5-670y5)5+(40x3y2)5+(800xy4)5

        (x5+2540y5)5+(x5-2540y5)5+(x5+1740y5)5+(x5-1740y5)5
        =(x5+2060y5)5+(x5-2060y5)5+(x5+340y5)5+(x5-340y5)5+(40x3y2)5+(800xy4)5

        (x5+2600y5)5+(x5-2600y5)5+(x5+1800y5)5+(x5-1800y5)5
        =(x5+2200y5)5+(x5-2200y5)5+(x5+200y5)5+(x5-200y5)5+(40x3y2)5+(800xy4)5

        (x5+2820y5)5+(x5-2820y5)5+(x5+2020y5)5+(x5-2020y5)5
        =(x5+2620y5)5+(x5-2620y5)5+(x5+220y5)5+(x5-220y5)5+(40x3y2)5+(800xy4)5

        (x5+2880y5)5+(x5-2880y5)5+(x5+2080y5)5+(x5-2080y5)5
        =(x5+2720y5)5+(x5-2720y5)5+(x5+320y5)5+(x5-320y5)5+(40x3y2)5+(800xy4)5




3. Method

(1+mx5)5+(1-mx5)5-(1+nx5)5-(1-nx5)5+(1+px5)5+(1-px5)5-(1+qx5)5-(1-qx5)5

=(10m4-10n4+10p4-10q4)x20+(20m2-20n2+20p2-20q2)x10................(1)

To do to make it to 5th powers the coefficient of x, m,n,p,q is decided. 

For example,first solution is (m,n,p,q)=(65,10,90,85),then we obtain

10m4-10n4+10p4-10q4=505
20m2-20n2+20p2-20q2=105..........................................(2)

Substitute (m,n,p,q)=(65,10,90,85) and (2) to (1), and obtain 
(1+65x5)5+(1-65x5)5-(1+10x5)5-(1-10x5)5+(1+90x5)5+(1-90x5)5-(1+85x5)5-(1-85x5)5-(10x2)5-(50x4)5

Substitute x=Y/X to above equation and simplify it,we obtain

(x5+65y5)5+(x5-65y5)5+(x5+90y5)5+(x5-90y5)5
=(x5+10y5)5+(x5-10y5)5+(x5+85y5)5+(x5-85y5)5+(10x3y2)5+(50xy4)5




4. Example

 1<=x<=5
 1<=y<=5   
         
(x,y)              
( 1 1)      665+      95+     915+     845 =     645+     115+     895+     865+     105+     505
( 1 2)    20815+    3195+   28815+   27195 =   20795+    3215+   28795+   27215+     405+    8005
( 1 3)   157965+   24295+  218715+  206545 =  157945+   24315+  218695+  206565+     905+   40505
( 1 4)   665615+  102395+  921615+  870395 =  665595+  102415+  921595+  870415+    1605+  128005
( 1 5)  2031265+  312495+ 2812515+ 2656245 = 2031245+  312515+ 2812495+ 2656265+    2505+  312505
( 2 1)      975-     225+    1225+     535 =     335+     425+     585+    1175+     805+    1005
( 2 3)   158275+   23985+  219025+  206235 =  157635+   24625+  218385+  206875+    7205+   81005
( 2 5)  2031575+  312185+ 2812825+ 2655935 = 2030935+  312825+ 2812185+ 2656575+   20005+  625005
( 3 1)     3085-    2335+    3335-    1585 =   -1785+    2535-    1535+    3285+    2705+    1505
( 3 2)    23235+     775+   31235+   24775 =   18375+    5635+   26375+   29635+   10805+   24005
( 3 4)   668035+   99975+  924035+  867975 =  663175+  104835+  919175+  872835+   43205+  384005
( 3 5)  2033685+  310075+ 2814935+ 2653825 = 2028825+  314935+ 2810075+ 2658685+   67505+  937505
( 4 1)    10895-   10145+   11145-    9395 =   -9595+   10345-    9345+   11095+    6405+    2005
( 4 3)   168195+   14065+  228945+  196315 =  147715+   34545+  208465+  216795+   57605+  162005
( 4 5)  2041495+  302265+ 2822745+ 2646015 = 2021015+  322745+ 2802265+ 2666495+  160005+ 1250005
( 5 1)     6385-    6235+    6435-    6085 =   -6125+    6275-    6075+    6425+    2505+     505
( 5 2)    10415-    5615+   12015-     815 =   -2095+    6895-     495+   11695+   10005+    8005
( 5 3)    37845-    1395+   49995+   35065 =   25345+   11115+   37495+   47565+   22505+   40505
( 5 4)   139375+   14235+  190575+  167835 =  126875+   26735+  178075+  180335+   40005+  128005