1.Introduction

We searched the rational solutions of x^4 + y^4 = cz^2 with 40000<=c<50000. 
Solution for c=40241,41729,42041,42577,44761,44489,46393,47273,48481,49129,49601 were not found.

Allan Macleod kindly sent me the solutions for n=40241,41729,42577,48481.(2019.1.30)

[c,x,y,z]=[40241, 2503031985486815770396487915526710051, 2032852404318689286676855110423714280, 37414052861558670280694787474240791471555283703833182802547259372432081]
[c,x,y,z]=[41729, 124286610518643769004868685, 193196116563208261972946732, 197746111311074958760722368632200978512138680260137]
[c,x,y,z]=[42577, 1905998019517942927420780438, 1998946671109168316972339691, 26171839524951319270285263375260563321287727060150169]
[c,x,y,z]=[48481, 261417536479490591265106148370, 328792258748106323956034663161, 580848954835950081166838653535047357437455153913218946281]

Now solution for c=42041,44761,44489,46393,47273,49129,49601 were not found.
It seems that x^4+y^4=cz^2 for above c has no rational solution.

(1). 1st step

     Reduce the search range by below condition.
     Condition:
           (a). c is squarefree and its odd divisors are congruent to 1 modulo 8. 
           (b). Y^2=X^3-4c^2X has a rational solution.
     If the two conditions are satisfied, we execute next step.

(2). 2nd step

     Set U=x/y, V=cz/y^2 then we get  V^2 = cU^4 + c.
     Find the rational solutions using Stoll's ratpoints.
     
(3). 3rd step
     
     Solving the simultaneous quadratic equation.
     For details, see below link.
     x^4 + y^4 = cz^2. 
     
(4). 4th step

     Set G=cUV, H=cU^2 then we get  G^2 = H^3 + c^2H.
     Find the rational solutions H such as H=cU^2. 
     
2. Search results

 40000<= c < 50000.

   c   [x, y, z]
[40241][62201038380925319211661934731411, 50516945511261879343874533695080, 23104534686452958355825133883355387769427305713953121567174001]
[40402][23134031, 81124821, 32850146536201]
[40466][1386864461, 400523545, 9594629395423801]
[40529][47884106995, 94535403212, 45829895778876158753]
[40673][9571, 7162, 520577]
[41201][73569820405376447050714139680, 88860834393283380433596987251, 47163180522582248011712261583734800146835294838585736401]
[41281][9274034994, 76424339795, 28749835812957727729]
[41537][373346, 240733, 740676689]
[41681][2124539, 2201660, 32442371281]
[41729][2978422931741565074765, 4629780645671074360108, 113561729268241507662463979388831085142057]
[42353][58, 31, 17]
[42577][44765906933742230016694, 46948978817417110575483, 14437238588237351499219837047678699265101561]
[43202][13, 11, 1]
[43826][188138245, 245061191, 332988246767369]
[43889][871788901, 352911140, 3676201190364097]
[44417][67453, 62468, 28441289]
[44641][1274, 495, 7769]
[44977][14, 9, 1]
[45169][17559, 4390, 1453537]
[45329][671195, 12702208, 757830471793]
[45377][809032546488538, 196807130059643, 3078032408141892223383858209]
[45569][20887, 11350, 2130937]
[46289][4555660, 6033397, 194760980273]
[46337][188455877, 55851466, 165624330954689]
[46769][11900, 827, 654817]
[46994][1534386335, 930724499, 11572263987187777]
[47137][14315901119990973351, 10828600952364725834, 1087550470520471495128193838642376649]
[47297][73909690612, 40307585473, 26205472288885364129]
[47713][2128, 327, 20737]
[47777][95543884, 55499113, 44076761573569]
[47858][739, 601, 2993]
[48161][1527190, 1046071, 11739262889]
[48353][1433161760472230591563, 1175898415074700228658, 11260112432938830112403415930283563761713]
[48481][5392164692961997303378770, 6781878648297401537840281, 247126929771366590426040898335600199576406599721]
[48722][2071, 1813, 24481]
[48817][49488, 39089, 13064809]
[48929][53, 50, 17]
[49057][41004910186248531, 19089951101510968, 7767649040413424783741974529561]
[49121][4216941176, 1858240315, 81733251229558441]
[49169][1021267, 727750, 5275307753]
[49297][13, 12, 1]
[49393][24690574805309599972, 13218300113163628089, 2853463320293659534467460741247333273]
[49586][2683845775, 4072107829, 81188336641763089]
[49618][1722419, 233799, 13320842473]
[49633][146956, 133209, 125462273]
[49778][581, 41, 1513]

3.Reference

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




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