1.Introduction

I searched the solutions of x^4 + y^4 = cz^2. 


(1). 1st step

     Search the solutions by brute force.
     Condition:
            x>y 
            x< 500000
     If no solution is found, we execute next step.

(2). 2nd step
     Set X=x^2/z, Y=y^2/z then we get X^2 + Y^2 = c.

     Example: c=929

     (X0,Y0)=(20,23) is a solution of X^2+Y^2=929.

     So, X = 2*(-10+10*k^2-23*k)/(1+k^2), Y = -(-23+23*k^2+40*k)/(1+k^2).

     Numerator of X and Y must be square.

     We solve the simultaneous equation { u^2 = -20+20*k^2-46*k, v^2 = 23-23*k^2-40*k }.

     Set k = -2*(2*s^2+63+22*s)/(s^2-20),and we get
     v^2 = -(185*s^4+66792*s^2+456748+6336*s^3+290224*s)/((s^2-20)^2).

     So,numerator of v^2 must be square.
     V^2 = -185*s^4-66792*s^2-456748-6336*s^3-290224*s.

     One solution is found s = -8774/1329.
     k = -8705123/20829128
     (X,Y) = (1179257440341124/509631739685513, 15488493205200625/509631739685513)

     Consequently,we get (x,y,z) = (34340318, 124452775, 509631739685513).
      





[1].Henri Cohen:Number Theory Volume 1:Tools and Diophantine Equations.


2. Search results

I searched the smallest solutions of (1) under the condition  c < 10000.
Search results.

Solutions of the case of c=2801,8081 and 9586 were not founded.

Allan MacLeod found the solutions of the case of c=7633,7762 and 8017.(2010.8.6)

c=7633:[37,  24,  17]

c=7762:[22147963,  11695113,  5780150291441]

c=8017:[80470366438,  24540786531,  72633376959449125289]


Allan MacLeod found the solutions of the case of c=2801,8081 and 9586.(2010.8.4)

c=2801:[270361295008966484650,90462483365506215707,1389752065073123173431809988480200625001]

c=8081:[130220386369,32892683140,189019921080221951129]

c=9586:[45488615038515,49266963181769,32576780589887658784973681]


    c   Solutions:[x, y, z]

   [2] [1, 1, 1]
  [17] [2, 1, 1]
  [82] [3, 1, 1]
  [97] [3, 2, 1]
 [113] [13, 8, 17]
 [193] [31, 18, 73]
 [257] [4, 1, 1]
 [274] [15, 13, 17]
 [337] [4, 3, 1]
 [433] [111, 26, 593]
 [514] [115, 93, 697]
 [577] [956, 477, 39209]
 [593] [68, 29, 193]
 [626] [5, 1, 1]
 [641] [5, 2, 1]
 [673] [21, 2, 17]
 [706] [5, 3, 1]
 [881] [5, 4, 1]
 [914] [935, 371, 29273]
 [929] [34340318, 124452775, 509631739685513]
[1153] [348, 143, 3617]
[1217] [324641, 121064, 3050147809]
[1297] [6, 1, 1]
[1409] [19943, 14230, 11889817 ]
[1522] [249, 131, 1649]
[1777] [227, 216, 1649]
[1873] [77, 6, 137]
[1889] [697800605, 223220836, 11261824700806937]
[1921] [6, 5, 1]
[2129] [28, 5, 17]
[2402] [7, 1, 1]
[2417] [7, 2, 1]
[2434] [327, 265, 2593]
[2482] [7, 3, 1]
[2498] [29, 11, 17]
[2642] [263, 181, 1489]
[2657] [7, 4, 1]
[2753] [77, 2, 113]
[2801] [270361295008966484650,90462483365506215707,1389752065073123173431809988480200625001]
[2833] [168236, 435543, 3603460457]
[2897] [233, 38, 1009]
[3026] [7, 5, 1]
[3121] [131, 120, 401]
[3137] [808, 221, 11689]
[3298] [179, 123, 617]
[3329] [6353933, 1210270, 700186946257]
[3457] [4869, 2602, 419329]
[3649] [675, 266, 7633]
[3697] [7, 6, 1]
[4001] [246260, 64499, 960999001]
[4097] [8, 1, 1]
[4129] [212941626, 183002665, 877267208280937]
[4177] [8, 3, 1]
[4226] [22447, 2615, 7751609]
[4289] [177119, 43940, 479924833]
[4481] [424, 95, 2689]
[4546] [853765, 833373, 14932473889]
[4561] [52731, 44410, 50477369]
[4721] [8, 5, 1]
[4817] [3706983931, 241535744, 197996264862688849]
[4993] [46537, 84, 30648977]
[5281] [30078190091, 7801782330, 12477455459557557841]
[5554] [119865, 26717, 193026593]
[5617] [348, 157, 1649]
[5666] [15851, 1205, 3337969]
[5729] [130, 71, 233]
[5906] [149, 25, 289]
[6002] [83, 17, 89]
[6353] [76504, 41117, 76432897]
[6449] [161855, 129044, 386544073]
[6481] [3600, 2027, 168881]
[6497] [8, 7, 1]
[6562] [9, 1, 1]
[6577] [9, 2, 1]
[6673] [76838, 11577, 72294193]
[6817] [9, 4, 1]
[6866] [967, 395, 11441]
[7057] [102375752833, 91975074786, 160331574499540446529,]
[7186] [9, 5, 1]
[7489] [232, 165, 697]
[7522] [513, 329, 3281]
[7537] [8651426491, 6400734342, 982842875526465769]
[7633] [37,  24,  17]
[7762] [22147963,  11695113,  5780150291441]
[8017] [80470366438,  24540786531,  72633376959449125289]
[8081] [130220386369,32892683140,189019921080221951129]
[8737] [320697, 293966, 1437139721]
[8753] [9133, 8518, 1181657]
[8882] [3251, 1241, 113329]
[8962] [9, 7, 1]
[9281] [25786472, 25997425, 9841667789249]
[9298] [41107, 5967, 17528033]
[9553] [1434, 929, 22817]
[9586] [45488615038515,49266963181769,32576780589887658784973681]
[9649] [143424, 89875, 224978993]
[9778] [41, 3, 17]
[9857] [94, 7, 89]






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