1.Introduction

We searched the rational solutions of x^4 - y^4 = cz^2. 
According to Cohen's book[1], this equation is related to congruent number problem.
That is to say, if c is congruent number, then x^4 - y^4 = cz^2 has a rational solution.
Generally, rational number c is congruent number if Y^2=X^3-c^2X has a rational solution.
Certainly, all c there is a solution for x^4 - y^4 = cz^2 is congruent number.

(1). 1st step

     Search the solutions by brute force.
     Condition:
            x>y 
            x< 1000
     If no solution is found, we execute next step.

(2). 2nd step

     Set U=x/y, V=cz/y^2 then we get  V^2 = cU^4 -c.
     This quartic equation is birationally equivalent to an elliptic curve below.
     Y^2 = X^3+6cX^2+16c^2X+16c^3
     Transformation is given, 
     U = (X+4c)/X, V = 4Yc/(X^2)
     X = 4c/(U-1), Y = 4cV/(U^2-2U+1).
     

2. Search results

 We show only smallest solution. 
 c < 200.

 c   [x, y, z]

[5] [3, 1, 4]
[6] [7, 1, 20]
[7] [4, 3, 5]
[13] [19, 17, 60]
[14] [79, 47, 1560]
[15] [2, 1, 1]
[20] [3, 1, 2]
[21] [31, 17, 200]
[22] [4999, 4801, 2058420]
[23] [156, 133, 3485]
[24] [7, 1, 10]
[28] [8, 6, 10]
[29] [99, 1, 1820]
[30] [17, 7, 52]
[31] [40, 9, 287]
[34] [5, 3, 4]
[37] [883, 881, 12180]
[38] [2201479,756479,780707760300]
[39] [5, 1, 4]
[41] [5, 4, 3]
[45] [9, 3, 12]
[46] [9839, 4273, 14017080]
[47] [3816, 1513, 2097655]
[52] [19, 17, 30]
[53] [55059,26737,404664260]
[54] [21, 3, 60]
[55] [13, 9, 20]
[56] [79, 47, 780]
[60] [4, 2, 2]
[61] [4723, 1361, 2846220]
[62] [2804352799,770844001,995924182207347120]
[63] [12, 9, 15]
[65] [3, 2, 1]
[69] [73199,56593,517119200]
[70] [73, 17, 636]
[71] [60, 11, 427]
[77] [15859663,15780913,4024693732800]
[78] [727, 623, 40620]
[79] [13000,12921,2950969]
[80] [3, 1, 1]
[84] [31, 17, 100]
[85] [11, 7, 12]
[86] [144967,78719,2165397780]
[87] [134, 13, 1925]
[88] [4999, 4801, 1029210]
[92] [312, 266, 6970]
[93] [2296111,1862911,411545395240]
[94] [213799199,183938207,3170212524438960]
[95] [39, 37, 68]
[96] [7, 1, 5]
[101] [220659, 201841, 2653270100]
[102] [2599, 2399, 350140]
[103] [93704884, 86901837, 441394452081515]
[109] [67, 31, 420]
[110] [119, 79, 1212]
[111] [7, 5, 4]
[112] [463, 113, 20220]
[116] [99, 1, 910]
[117] [19, 17, 20]
[118] [540376768995259399, 485081984490254399, 15918195554075977509065598561353100]
[119] [12, 5, 13]
[120] [17, 7, 26]
[124] [80, 18, 574]
[125] [15, 5, 20]
[126] [79, 47, 520]
[127] [17501923504, 17467450497, 2409099421727351105]
[133] [118514415967, 28768756817, 1215797536433565897240]
[134] [4405580448007, 4287484391999, 538083593367127708669860]
[135] [698, 671, 16021]
[136] [5, 3, 2]
[137] [81, 17, 560]
[138] [623, 527, 23080]
[141] [2399, 2207, 258160]
[142] [25777019425602293039, 22549204281387444239, 35895171275697097635053683913361268920]
[143] [318, 43, 8455]
[145] [7, 3, 4]
[148] [883, 881, 6090]
[149] [387, 191, 11900]
[150] [7, 1, 4]
[151] [340, 189, 8947]
[152] [2201479, 756479, 390353880150]
[154] [113, 41, 1020]
[156] [5, 1, 2]
[157] [780871468723, 526771095761, 43333111387429522619220]
[158] [1146128756959, 634784142241, 99467085094604485740720]
[159] [182, 107, 2465]
[161] [15, 8, 17]
[164] [33, 31, 40]
[165] [487, 217, 18096]
[166] [67800685274972167, 8111876915714881, 356754634811763605742002426197020]
[167] [339444, 58717, 8912171435]
[173] [69084044403, 6056633521, 362843735226119908540]
[174] [833, 617, 43980]
[175] [4, 3, 1]
[180] [49, 31, 164]
[181] [65539, 43039, 288057900]
[182] [129673, 104977, 941424540]
[183] [4279, 4171, 421980]
[184] [9839, 4273, 7008540]
[188] [7632, 3026, 4195310]
[189] [93, 51, 600]
[190] [199, 161, 2172]
[191] [636182040, 620449561, 9040925636810543]
[194] [13, 5, 12]
[197] [45773379, 43444703, 64808747718260]
[198] [4999, 4801, 686140]
[199] [3732820, 523149, 987560631301]


3.Reference

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




HOME