1.Introduction

We searched the rational solutions of x^4 + y^4 = cz^2 with 30000<=c<40000. 
Solution for c= 31529, 34241, 35513, 36121 were not found.
It seems that x^4+y^4=cz^2 for c=31529, 34241, 35513, 36121 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

 30000<= c < 40000.

   c   [x, y, z]

[30146][33853565, 26099339, 7678658763889]
[30241][1094827688962069071032, 11502113624434063607685, 760807976224816893205679957597623451356169]
[30497][54946, 1987, 17287969]
[30593][1810755921302, 4644435874753, 124742669146931367583273]
[30689][291625833938859800, 61642005364329521, 485952314701655886683185094188273]
[30722][72040057, 136965343, 111047932437721]
[30962][13, 7, 1]
[31138][5187, 373, 152473]
[31153][44060904, 590939813, 1978532890669217]
[31474][5087050799991411645, 4517612217885243703, 185771027311426658711755321652821313]
[31522][659928007, 3372030207, 64090601122973521]
[31649][11230646093630320, 8690770347806489, 826372392198793590658441073953]
[32114][55, 19, 17]
[32369][41851700395, 7682261266, 9741099009883160713]
[32689][843030, 3766457, 78561456097]
[32834][66598817125, 416287431869, 956680459476157582913]
[33073][140628, 102661, 123222817]
[33346][2784620748462106437669513845, 1638122516458424919269694843, 44933807473197743491203173051009907087853462058243641]
[33377][21401, 8752, 2541761]
[33601][17762, 4155, 1723681]
[33617][10628, 6523, 658321]
[33857][294439737350371, 1222741441680184, 8139054079715514107377671209]
[33937][17971884757778173047, 5388097130081651108, 1760345231189667000671711969104451689]
[34018][323461, 181677, 594827593]
[34033][914, 783, 5617]
[34129][368629948631261, 498937033521540, 1535192449249886922114311873]
[34289][684340, 205133, 2539291217]
[34513][5032052, 9260259, 481291829377]
[34802][1277, 839, 9521]
[34913][815174, 40187, 3556381457]
[35122][13, 9, 1]
[35281][52460530, 21459501, 14855638490761]
[35377][12, 11, 1]
[36017][59017, 1732, 18352729]
[36097][306568, 211839, 548170369]
[36209][27290038505, 27651930676, 5609339201726821817]
[36241][57069067205871, 615377052199130, 1989292150596773798323976929]
[36529][1302897142940, 1711343941899, 17711400784534820888513]
[36737][316, 29, 521]
[38033][15854, 14813, 1710857]
[38321][2195, 932, 25009]
[38417][14, 1, 1]
[38449][16421290, 50099163, 12873911667017]
[38497][14, 3, 1]
[38546][4891859202807035, 5214084295928791, 184475984160779949373309864321]
[38561][13, 10, 1]
[38737][1485239942, 2121684489, 25470287640345649]
[38897][641, 428, 2281]
[38977][5195280919, 452097426, 136718195824417369]
[39041][14, 5, 1]
[39089][16310, 5153, 1352177]

3.Reference

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




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