1.Introduction

We searched the rational solutions of x^4 + y^4 = cz^2 with 10000<=c<20000. 
According to Cohen's book[1], this equation is related to congruent number problem.
That is to say, if x^4 + y^4 = cz^2 has a rational solution, then 2c is congruent number.
Generally, rational number 2c is congruent number if Y^2=X^3-4c^2X has a rational solution.
Hence if Y^2=X^3-4c^2X has no rational solution, then x^4 + y^4 = cz^2 has no rational solution.
However if Y^2=X^3-4c^2X has a rational solution, x^4 + y^4 = cz^2 does not always have solutions.
If c=a^4+b^4 then x^4 + y^4 = cz^2 has a obvious solution (x,y,z)=(a,b,1).

Solvability  for c=15929.(2019.1.22)
G^2=H^3+c^2H has two generators {p1,p2} below.
p1(H,G)=(3748/9, -8779690/27)
p2(H,G)=(840766187133913/14592156804, -25293318443821421073125/1762703357609592)
No rational solution was found such as H=cu^2 with H<1000 digits.
It seems that x^4+y^4=cz^2 for c=15929 has no rational solution.

Allan Macleod kindly sent me the solutions for n=13249,14177,18754,19121.(2019.1.19)

c = 13249
[x,y,z]=[339785965139155902176647645, 819666681242379201079841406, 5922466980820738452264320674855584519137678323781777]

c = 14177
[x,y,z]=[184385149355462843474, 400010781821191941077, 1373851070882230186662013324561959767089]

c = 18754
[x,y,z]=[51084516112944037681365090, 63082257195104107191252914, 34749175218582865758706716965929522155529576766948]

c = 19121
[x,y,z]=[4867164372975929792674720308425, 9529663388363483868912823443964, 678726051910476397729733431216635870348420041828061282457561]

Solution for c= 15929 was not found.

(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

 10000<= c < 20000.

   c   [x, y, z]

[10001][10, 1, 1]
[10081][10, 3, 1]
[10193][29203252442, 12633982279, 8593851301456209233]
[10321][247123586, 725480895, 5215489715390561]
[10513][326050336, 41396067, 1036960914482657]
[10529][14605, 3406, 2081857]
[10546][1214325, 896471, 16353155369]
[10657][9, 8, 1]
[10898][16666997, 34570559, 11753442583177]
[11057][19661948679688, 119261652996523, 135314193835001765484786881]
[11138][8581, 4123, 716057]
[11314][2059, 15, 39857]
[11521][419, 150, 1649]
[11617][38196, 1991, 13536001]
[11633][2828055988, 2924098069, 108550815918586273]
[11698][12282567, 17888237, 3270870525553]
[11969][1280, 143, 14977]
[12146][9775, 3269, 872401]
[12193][43, 18, 17]
[12401][10, 7, 1]
[12433][26587, 7218, 6356633]
[12497][26476139921, 38441797684, 14631016564180829081]
[12577][549250606032, 1345137454039, 16356813842960225922689]
[12721][90, 77, 89]
[12833][148279, 115708, 227238337]
[13009][6634786, 3706365, 404306833193]
[13058][620947, 247523, 3416531353]
[13249][25646159343282957368605, 61866305475309774404094, 33739329522613076119150741475850213308309777]
[13313][811, 394, 5857]
[13729][529831519382750, 221874843545943, 2432390716332639185676106537]
[13841][1540136, 907505, 21342713401]
[13921][109919370100, 513016978743, 2232983013827130164641]
[14002][521287, 166029, 2308244689]
[14177][13005935624988562, 28215474488339701, 6835510863882519612278708995441]
[14369][8635, 1352, 622217]
[14561][563521, 402370, 2953917689]
[14642][11, 1, 1]
[14657][11, 2, 1]
[14722][11, 3, 1]
[14834][244745, 152747, 527802553]
[14897][11, 4, 1]
[14977][3227, 2934, 110401]
[15122][113123, 98801, 130883681]
[15266][11, 5, 1]
[15586][103031, 30735, 85365281]
[15889][1277, 30, 12937]
[15937][11, 6, 1]
[16033][15051, 10622, 1998673]
[16097][920546, 89093, 6679401809]
[16273][5423266, 3644589, 252984844457]
[16561][10, 9, 1]
[16609][519, 200, 2113]
[16673][287281508639, 593001975788, 2797360379111075415857]
[17042][11, 7, 1]
[17153][275369, 144544, 600550417]
[17218][1025497, 435111, 8143353377]
[17393][27254482, 41242643, 14073694314233]
[17489][199017994658290, 23943523719473, 299535078859546163786537537]
[17921][19482547427, 80901619850, 48973641909171807089]
[18049][2131569, 4792790, 174294599177]
[18098][57097, 2491, 24233273]
[18353][190579, 140036, 304681457]
[18514][136434265, 43560549, 137512326667297]
[18562][66814711703, 187963319289, 261380637706590128641]
[18737][11, 8, 1]
[18754][2723926421720381661585, 3363669467585800746041, 98799939262277677257032038535499607172153]
[18994][51071, 16515, 19028393]
[19042][590359, 194409, 2540474329]
[19121][254545493069187270157142425, 498387290851079120805021884, 1856407508378454556033070615894366383104407054208121]
[19202][984157223, 18285431, 6989655347315041]
[19601][514850, 185413, 1909169369]
[19777][20616, 12739, 3235049]
[19793][312283, 11572, 693172433]

3.Reference

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




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