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
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