1.Introduction

We searched the rational solutions of x^4 + y^4 = cz^2 with 20000<=c<30000. 
Solution for c= 22873 was not found.
It seems that x^4+y^4=cz^2 for c=22873 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

 20000<= c < 30000.

   c   [x, y, z]

[20177][666688217, 167261908, 3135276629353961]
[20513][431, 8, 1297]
[20674][45, 37, 17]
[20689][521682, 473065, 2449637297]
[20737][12, 1, 1]
[20897][16436, 4091, 1872329]
[20929][73814321410950, 123922432029989, 112634648791780939008710177]
[20978][529, 509, 2633]
[21202][11, 9, 1]
[21361][12, 5, 1]
[21649][50, 9, 17]
[21713][118168, 34477, 95105953]
[21809][733250, 6598283, 294833694313]
[22193][28937, 20522, 6291713]
[22738][497, 459, 2153]
[22961][16478772701028922878334, 26055013658577090485495, 4825220768705034906739024700715810652454401]
[23057][6536870975888, 2115420736579, 282948256327999250119841]
[23137][12, 7, 1]
[23377][207, 56, 281]
[23537][762029, 1127036, 9103579409]
[23633][61369, 19412, 24620777]
[24113][5152002504401, 12041254452898, 949239674584743923726353]
[24322][1864123, 436371, 22315188449]
[24482][60142673, 30202279, 23841320057761]
[24593][21518, 2033, 2952673]
[24641][11, 10, 1]
[24929][35721970665588362, 9745914518850895, -8104349716094004792649177404697]
[25169][89968, 20555, 51089777]
[25313][5396, 3437, 197497]
[25409][2480, 991, 39073]
[25618][9460692957, 12543378359, 1130935645986160697]
[25666][877, 75, 4801]
[25778][6091, 3289, 240697]
[25793][1336, 737, 11617]
[25841][247070, 150577, 405087601]
[26881][122860, 44013, 92820841]
[26882][1076531, 1661323, 18257412529]
[26993][1548164723, 303082736644, 559109437231701947153]
[27026][61205947745, 881255581, 22787484038008077361]
[27217][38673, 25958, 9943169]
[27361][198, 85, 241]
[27682][22736182452039, 19549931882953, 3863960161067434276491161]
[27809][395, 92, 937]
[27842][11317, 437, 767561]
[28081][427786315880286, 1585529174108405, -15041455375895531152970589481]
[28178][20191, 12053, 2578217]
[28562][13, 1, 1]
[28642][13, 3, 1]
[28753][242604427, 10051608, 347101508219377]
[28817][13, 4, 1]
[28898][13707109, 6954973, 1141284615193]
[29186][13, 5, 1]
[29473][54, 11, 17]
[29537][362, 131, 769]
[29857][13, 6, 1]
[29858][69523, 28411, 28359577]

3.Reference

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




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