1.Introduction

A. Bremner and J.W. Jones showed numeric solutions of x^4+mx^2y^2+y^4=z^2 for |m|<=200.
We show numeric solutions of x^4+mx^2y^2+y^4=z^2 for 200<|m|<1000.
It's seems that height of solutions are sometimes very large when m<0.
In particular, 4 descent method is required for m=-615,-675,-725,-771,-813,-825,-855,-939,-969.
However, n mentioned above may not be the smallest solution.

Furthermore, we show that x^4+mx^2y^2+y^4=z^2 has infinitely many integer solutions where m={2t^2+6,2t^2-6}.
Allan J. MacLeod[3] already had given the solutions for |m|<10000.


2.Thorem

Diophantine equation x^4+mx^2y^2+y^4=z^2 has infinitely many integer solutions.

m=2t^2+6 and 2t^2-6.
t is arbitrary.

Proof.

x^4+mx^2y^2+y^4=z^2................................................(1)

Let X=x/y, Y=z/y.
Y^2 = X^4+mX^2+1...................................................(2)

According to Allan J. MacLeod[2], equation (2) can be transformed to elliptic curve (3).

S^2 = T^3-2mT^2+(m^2-4)T...........................................(3)
X=S/(2T), Y=(2X^2+m-T)/2

Case 1: m = 2t^2+6

Let (T,m)=(n,2n-2) and n = t^2+4, then we obtain S^2=t^2(t^2+4)^2.

Hence S=t(t^2+4).

We know elliptic curve (3) has a point P(T,S)=(t^2+4, t(t^2+4)).

[m,x,y,z]=[2t^2+6, t, 2, 3t^2+4]

For example k=2.

[m,x,y,z]=[2t^2+6, -(t^4+24t^2+16)(t^4-24t^2-48)t, 2(5t^4+8t^2+16)(7t^4+8t^2-16),
          (3t^2+4)(33t^16+96t^14+20416t^12+109056t^10+206336t^8+172032t^6+245760t^4+65536+393216t^2)]

This point P is of infinite order, and the multiples kP, k = 2, 3, ...give infinitely many points.



Case 2: m = 2t^2-6

Let (T,m)=(n,2n+2) and n = t^2-4, then we obtain S^2=t^2(t^2-4)^2.

Hence S=t(t^2-4).

We know elliptic curve (3) has a point P(T,S)=(t^2-4, t(t^2-4)).

[m,x,y,z]=[2t^2-6, t, 2, 3t^2-4]

For example k=2.

[m,x,y,z]=[2t^2-6, -(t^4+24t^2-48)(t^2-4t-4)(t^2+4t-4)t, 2(7t^4-8t^2-16)(5t^4-8t^2+16),
          (3t^2-4)(65536-393216t^2+245760t^4-172032t^6+206336t^8-109056t^10+20416t^12-96t^14+33t^16)]
          
This point P is of infinite order, and the multiples kP, k = 2, 3, ...give infinitely many points.


3.Method

We searched the rational solutions of elliptic curve (3).

Step1: Brute force(Pari-gp)
Step2: Search rank and generator(Sage: 2 descent)
Step3: Search rational point(Magma online: 4 descent)
       m=-615,-675,-725,-771,-813,-825,-855,-939,-969
       
200<|m|<1000


4.Results

Smallest soutions are shown.

[m, 0, 0, 0]: No solution.

[m, x, y, z]

[201, 22360, 393, 515257999]
[202, 0, 0, 0]
[203, 55, 8, 6961]
[204, 525, 442, 3331511]
[205, 0, 0, 0]
[206, 2, 1, 29]
[207, 0, 0, 0]
[208, 12, 11, 1913]
[209, 0, 0, 0]
[210, 85, 78, 96541]
[211, 20, 3, 959]
[212, 6, 5, 439]
[213, 987, 44, 1162207]
[214, 0, 0, 0]
[215, 0, 0, 0]
[216, 70, 3, 5791]
[217, 0, 0, 0]
[218, 0, 0, 0]
[219, 8, 3, 361]
[220, 0, 0, 0]
[221, 56, 3, 4009]
[222, 0, 0, 0]
[223, 0, 0, 0]
[224, 0, 0, 0]
[225, 0, 0, 0]
[226, 3, 1, 46]
[227, 7, 4, 425]
[228, 0, 0, 0]
[229, 9, 4, 551]
[230, 35, 2, 1621]
[231, 0, 0, 0]
[232, 4, 1, 63]
[233, 12, 1, 233]
[234, 6, 5, 461]
[235, 0, 0, 0]
[236, 2, 1, 31]
[237, 54793, 7752, 7195561487]
[238, 14, 3, 677]
[239, 9, 1, 161]
[240, 0, 0, 0]
[241, 0, 0, 0]
[242, 14, 1, 293]
[243, 0, 0, 0]
[244, 15, 8, 1889]
[245, 95, 84, 125431]
[246, 0, 0, 0]
[247, 9, 1, 163]
[248, 4, 1, 65]
[249, 85, 84, 113119]
[250, 0, 0, 0]
[251, 4, 3, 191]
[252, 14, 3, 695]
[253, 0, 0, 0]
[254, 0, 0, 0]
[255, 5, 3, 241]
[256, 0, 0, 0]
[257, 9, 4, 583]
[258, 0, 0, 0]
[259, 0, 0, 0]
[260, 0, 0, 0]
[261, 48, 1, 2431]
[262, 0, 0, 0]
[263, 7, 4, 457]
[264, 0, 0, 0]
[265, 1677, 1480, 40560329]
[266, 0, 0, 0]
[267, 0, 0, 0]
[268, 2, 1, 33]
[269, 56, 5, 5561]
[270, 35, 3, 2116]
[271, 0, 0, 0]
[272, 0, 0, 0]
[273, 21, 16, 5575]
[274, 0, 0, 0]
[275, 180, 11, 46129]
[276, 12, 11, 2201]
[277, 0, 0, 0]
[278, 0, 0, 0]
[279, 0, 0, 0]
[280, 0, 0, 0]
[281, 0, 0, 0]
[282, 6, 1, 107]
[283, 72, 11, 14297]
[284, 7, 2, 241]
[286, 5, 2, 171]
[287, 0, 0, 0]
[288, 6, 5, 511]
[289, 0, 0, 0]
[290, 0, 0, 0]
[291, 689, 168, 2031041]
[292, 3, 2, 103]
[293, 0, 0, 0]
[294, 6, 1, 109]
[295, 5, 3, 259]
[296, 10, 1, 199]
[297, 0, 0, 0]
[298, 0, 0, 0]
[299, 4, 1, 71]
[300, 7072270, 896427, 120665512053671]
[301, 4, 3, 209]
[302, 2, 1, 35]
[303, 3, 1, 53]
[304, 10, 1, 201]
[305, 2372329, 27312740, 1355370172441991]
[306, 15, 2, 571]
[307, 13108, 1005, 287750689]
[308, 0, 0, 0]
[309, 0, 0, 0]
[310, 0, 0, 0]
[311, 15, 11, 2921]
[312, 0, 0, 0]
[313, 16, 3, 887]
[314, 5, 2, 179]
[315, 1561122685, 2464838232, 68606717627773117801]
[316, 0, 0, 0]
[317, 4, 1, 73]
[318, 0, 0, 0]
[319, 7705, 693, 112337299]
[320, 5, 4, 359]
[321, 0, 0, 0]
[322, 0, 0, 0]
[323, 16, 1, 385]
[324, 0, 0, 0]
[325, 0, 0, 0]
[326, 35, 4, 2809]
[327, 3, 1, 55]
[328, 104, 57, 107953]
[329, 38613, 6020, 4472268319]
[330, 1819, 660, 22062761]
[331, 8, 1, 159]
[332, 13, 2, 503]
[334, 0, 0, 0]
[335, 0, 0, 0]
[336, 10, 3, 559]
[337, 0, 0, 0]
[338, 2, 1, 37]
[339, 0, 0, 0]
[340, 0, 0, 0]
[341, 8, 1, 161]
[342, 36, 1, 1457]
[343, 4, 3, 223]
[344, 13, 2, 511]
[345, 0, 0, 0]
[346, 0, 0, 0]
[347, 1292, 35, 1869761]
[348, 0, 0, 0]
[349, 11, 4, 831]
[350, 0, 0, 0]
[351, 0, 0, 0]
[352, 3, 2, 113]
[353, 0, 0, 0]
[354, 0, 0, 0]
[355, 211888, 8025, 55155565169]
[356, 0, 0, 0]
[358, 7, 6, 797]
[359, 1, 1, 19]
[360, 70, 9, 12919]
[361, 287, 120, 659681]
[362, 0, 0, 0]
[363, 1067, 696, 14203033]
[364, 0, 0, 0]
[365, 0, 0, 0]
[366, 12, 1, 271]
[367, 5, 1, 99]
[368, 55, 14, 15079]
[369, 57333, 46592, 51464191879]
[370, 195, 8, 48439]
[371, 155, 104, 311609]
[372, 13, 12, 3017]
[373, 0, 0, 0]
[374, 4, 1, 79]
[375, 1547, 60, 2993041]
[376, 2, 1, 39]
[377, 11, 4, 863]
[378, 0, 0, 0]
[379, 1196, 495, 11616409]
[380, 5, 4, 391]
[381, 0, 0, 0]
[382, 0, 0, 0]
[383, 5, 1, 101]
[384, 20328, 8303, 3333892129]
[385, 22240, 2233, 1092794289]
[386, 7, 1, 146]
[387, 9, 4, 713]
[388, 20, 9, 3569]
[389, 208, 77, 318889]
[390, 0, 0, 0]
[391, 1353, 551, 14857699]
[392, 20, 7, 2801]
[393, 0, 0, 0]
[394, 4, 1, 81]
[395, 0, 0, 0]
[396, 3581218, 3651459, 260879255728919]
[397, 220, 51, 228751]
[398, 1, 1, 20]
[399, 0, 0, 0]
[400, 0, 0, 0]
[401, 4, 3, 241]
[402, 18, 1, 485]
[403, 0, 0, 0]
[404, 3, 2, 121]
[405, 0, 0, 0]
[406, 105, 44, 93761]
[407, 0, 0, 0]
[408, 18, 1, 487]
[409, 60, 1, 3799]
[410, 0, 0, 0]
[411, 0, 0, 0]
[412, 6, 1, 127]
[413, 0, 0, 0]
[414, 102, 7, 17869]
[415, 12, 5, 1231]
[416, 2, 1, 41]
[417, 18902521, 879710244, 845111120890322663]
[418, 3, 1, 62]
[419, 0, 0, 0]
[420, 0, 0, 0]
[421, 1441, 140, 4631031]
[422, 0, 0, 0]
[423, 9, 4, 745]
[424, 0, 0, 0]
[425, 639, 160, 2147071]
[426, 7, 6, 869]
[427, 0, 0, 0]
[428, 12, 1, 287]
[429, 48, 5, 5479]
[430, 0, 0, 0]
[431, 0, 0, 0]
[432, 8, 3, 503]
[433, 0, 0, 0]
[434, 0, 0, 0]
[435, 0, 0, 0]
[436, 12, 1, 289]
[437, 8, 5, 839]
[438, 987, 46, 1360837]
[439, 0, 0, 0]
[440, 70, 11, 16879]
[441, 0, 0, 0]
[442, 30, 1, 1099]
[443, 55, 16, 18769]
[444, 15, 2, 671]
[445, 110399710987580, 7018225031163, 20388703090506918538461820319]
[446, 3, 1, 64]
[447, 72, 5, 9209]
[448, 0, 0, 0]
[449, 0, 0, 0]
[450, 35, 9, 6794]
[451, 0, 0, 0]
[452, 7, 6, 895]
[453, 21, 20, 8959]
[454, 371, 36, 316121]
[455, 15, 13, 4169]
[456, 15, 2, 679]
[457, 4, 1, 87]
[458, 2, 1, 43]
[459, 29170577515, 279916380072, 191683640905206048297241]
[460, 145, 36, 113921]
[461, 28808, 13969, 8682258881]
[462, 0, 0, 0]
[463, 0, 0, 0]
[464, 8, 3, 521]
[465, 0, 0, 0]
[466, 0, 0, 0]
[467, 0, 0, 0]
[468, 0, 0, 0]
[469, 2119, 84, 5917831]
[470, 0, 0, 0]
[471, 0, 0, 0]
[472, 63, 8, 11647]
[473, 0, 0, 0]
[474, 3, 2, 131]
[475, 325, 4, 109359]
[476, 9, 2, 401]
[477, 2376, 247, 14005775]
[478, 4, 3, 263]
[479, 4, 1, 89]
[480, 14662151, 9190680, 2961358713802801]
[481, 75, 8, 14311]
[482, 1, 1, 22]
[483, 535, 312, 3680881]
[484, 5, 4, 441]
[485, 0, 0, 0]
[486, 35, 6, 4789]
[487, 0, 0, 0]
[488, 0, 0, 0]
[489, 0, 0, 0]
[490, 0, 0, 0]
[491, 8, 5, 889]
[492, 0, 0, 0]
[493, 20, 3, 1391]
[494, 28, 5, 3209]
[495, 0, 0, 0]
[496, 9, 2, 409]
[497, 20, 1, 599]
[498, 0, 0, 0]
[499, 551, 24, 423599]
[500, 0, 0, 0]
[501, 0, 0, 0]
[502, 2, 1, 45]
[503, 13, 1, 337]
[504, 0, 0, 0]
[505, 110425, 1056, 12472074511]
[506, 8, 1, 191]
[507, 20, 3, 1409]
[508, 0, 0, 0]
[509, 0, 0, 0]
[510, 0, 0, 0]
[511, 13, 1, 339]
[512, 44, 1, 2177]
[513, 0, 0, 0]
[514, 0, 0, 0]
[515, 1813, 220, 9630031]
[516, 0, 0, 0]
[517, 0, 0, 0]
[518, 8, 1, 193]
[519, 7, 3, 481]
[520, 10, 1, 249]
[521, 13, 4, 1199]
[522, 0, 0, 0]
[523, 72, 13, 22025]
[524, 70, 19, 30839]
[525, 1010295, 7952, 1037162174929]
[526, 0, 0, 0]
[527, 0, 0, 0]
[528, 7, 6, 967]
[529, 0, 0, 0]
[530, 10, 1, 251]
[531, 464, 75, 830329]
[532, 6, 1, 143]
[533, 836, 63, 1402487]
[534, 3, 2, 139]
[535, 0, 0, 0]
[536, 70, 13, 21631]
[537, 0, 0, 0]
[538, 0, 0, 0]
[539, 0, 0, 0]
[540, 11570, 9177, 2472414671]
[541, 325, 12, 139231]
[542, 0, 0, 0]
[544, 17, 12, 4769]
[545, 383799, 21340, 241364622151]
[546, 4, 3, 281]
[547, 21, 20, 9841]
[548, 2, 1, 47]
[549, 220, 57, 297799]
[550, 28470, 118643, 80460628949]
[551, 3, 1, 71]
[552, 0, 0, 0]
[553, 6765, 112, 49111319]
[554, 0, 0, 0]
[555, 0, 0, 0]
[556, 0, 0, 0]
[557, 56, 11, 14873]
[558, 22, 21, 10933]
[559, 7, 3, 499]
[560, 4620, 47, 21954209]
[561, 1037, 156, 3979759]
[562, 0, 0, 0]
[563, 0, 0, 0]
[564, 28085, 3192, 2270445689]
[565, 260, 11, 95871]
[566, 99, 8, 21239]
[567, 72, 7, 13073]
[568, 69, 28, 46297]
[569, 0, 0, 0]
[570, 0, 0, 0]
[571, 0, 0, 0]
[572, 4, 1, 97]
[573, 2596, 69, 7987615]
[574, 1, 1, 24]
[575, 7337, 95, 56366569]
[576, 0, 0, 0]
[577, 0, 0, 0]
[578, 0, 0, 0]
[579, 0, 0, 0]
[580, 0, 0, 0]
[581, 0, 0, 0]
[582, 295, 156, 1113889]
[583, 3, 1, 73]
[584, 14, 1, 391]
[585, 1631178545, 250045068, 10217728470787117751]
[586, 0, 0, 0]
[587, 11, 4, 1073]
[588, 0, 0, 0]
[590, 5, 1, 124]
[591, 39, 7, 6809]
[592, 14, 1, 393]
[593, 48, 11, 13063]
[594, 3570, 871, 76852309]
[595, 0, 0, 0]
[596, 2, 1, 49]
[597, 48, 7, 8527]
[598, 22, 3, 1685]
[599, 209, 21, 115961]
[600, 570, 187, 2631281]
[601, 2405, 304, 18834009]
[602, 22, 1, 725]
[603, 0, 0, 0]
[604, 133, 78, 255641]
[605, 0, 0, 0]
[606, 645, 238, 3802219]
[607, 0, 0, 0]
[608, 22, 1, 727]
[609, 164339593, 2090592, 28307075171710639]
[610, 5, 1, 126]
[611, 115, 4, 17441]
[612, 22, 3, 1703]
[613, 1793, 1596, 70969207]
[614, 3, 2, 149]
[615, 0, 0, 0]
[616, 0, 0, 0]
[617, 0, 0, 0]
[618, 15, 2, 779]
[619, 0, 0, 0]
[620, 0, 0, 0]
[621, 2120, 147, 8972809]
[622, 0, 0, 0]
[623, 1, 1, 25]
[624, 5117, 228, 39178249]
[625, 432, 205, 2222249]
[626, 14, 3, 1069]
[627, 0, 0, 0]
[628, 20, 11, 5529]
[629, 0, 0, 0]
[630, 0, 0, 0]
[631, 0, 0, 0]
[632, 0, 0, 0]
[633, 2728, 69, 8821063]
[634, 0, 0, 0]
[635, 12, 5, 1519]
[636, 0, 0, 0]
[637, 0, 0, 0]
[638, 0, 0, 0]
[639, 0, 0, 0]
[640, 15, 14, 5321]
[641, 0, 0, 0]
[642, 9, 1, 242]
[643, 1555560, 1378663, 54468380789969]
[644, 39, 4, 4241]
[645, 0, 0, 0]
[646, 2, 1, 51]
[647, 4, 1, 103]
[648, 14, 3, 1087]
[649, 87, 8, 19279]
[650, 0, 0, 0]
[651, 0, 0, 0]
[652, 22, 21, 11815]
[653, 20, 1, 649]
[654, 9, 1, 244]
[655, 1771, 425, 19517691]
[656, 7, 4, 719]
[657, 15729895, 328538652, 170871515215480871]
[658, 36, 7, 6593]
[659, 13920, 1853, 689928841]
[660, 0, 0, 0]
[661, 15, 4, 1559]
[662, 0, 0, 0]
[663, 0, 0, 0]
[664, 0, 0, 0]
[665, 132, 85, 289951]
[666, 462, 391, 4669211]
[667, 0, 0, 0]
[668, 0, 0, 0]
[669, 0, 0, 0]
[670, 0, 0, 0]
[671, 15, 1, 449]
[672, 0, 0, 0]
[673, 4, 1, 105]
[674, 0, 0, 0]
[675, 1735767644157175, 3570709556699688,161558813354377892843971291343281]
[676, 0, 0, 0]
[677, 0, 0, 0]
[678, 4, 3, 313]
[679, 15, 1, 451]
[680, 265, 28, 205841]
[681, 27401, 7848, 5662097521]
[682, 3, 2, 157]
[683, 39032, 6649, 6951605137]
[684, 6, 1, 161]
[685, 5266894297329501420, 11720797319734042243,1621753441134884206935501408338531826799]
[686, 0, 0, 0]
[687, 0, 0, 0]
[688, 19, 12, 5993]
[689, 15, 4, 1591]
[690, 0, 0, 0]
[691, 0, 0, 0]
[692, 75, 2, 6871]
[694, 135, 14, 53021]
[695, 0, 0, 0]
[696, 6428523, 16914830, 2883217620516271]
[697, 0, 0, 0]
[698, 2, 1, 53]
[699, 0, 0, 0]
[700, 0, 0, 0]
[701, 28, 1, 1079]
[702, 3, 1, 80]
[703, 0, 0, 0]
[704, 0, 0, 0]
[705, 15, 4, 1609]
[706, 0, 0, 0]
[707, 1441, 20, 2213369]
[708, 0, 0, 0]
[709, 0, 0, 0]
[710, 35, 8, 7561]
[711, 273, 11, 109391]
[712, 0, 0, 0]
[713, 8, 1, 223]
[714, 74315, 13464, 27301240121]
[715, 84216, 82315, 185623861031]
[716, 52, 1, 3041]
[717, 24, 1, 863]
[718, 0, 0, 0]
[719, 0, 0, 0]
[720, 1241, 450, 15065081]
[721, 252294120, 127229641, 864411019029220081]
[722, 55, 26, 38549]
[723, 24, 1, 865]
[724, 8, 3, 649]
[725, 130885409185089, 14299676898080, 53227409569506994024559137921]
[726, 138, 95, 353869]
[727, 1, 1, 27]
[728, 5, 2, 271]
[729, 6279, 376, 74951761]
[730, 138, 85, 317581]
[731, 0, 0, 0]
[732, 0, 0, 0]
[733, 12, 5, 1631]
[734, 0, 0, 0]
[735, 2945, 117, 12746939]
[736, 11, 2, 609]
[737, 2948, 247, 21593927]
[738, 3, 1, 82]
[739, 0, 0, 0]
[740, 0, 0, 0]
[741, 0, 0, 0]
[742, 0, 0, 0]
[743, 7, 1, 197]
[744, 0, 0, 0]
[745, 2717, 1480, 110026089]
[746, 21, 2, 1229]
[747, 171700, 1792251, 9003213562001]
[748, 0, 0, 0]
[749, 0, 0, 0]
[750, 216, 95, 563969]
[751, 12, 1, 359]
[752, 2, 1, 55]
[753, 100114369722885, 23665005843688, 65783494878778085380492562119]
[754, 4, 1, 111]
[755, 0, 0, 0]
[756, 135, 2, 19679]
[757, 413, 180, 2052719]
[758, 0, 0, 0]
[759, 0, 0, 0]
[760, 0, 0, 0]
[761, 12, 1, 361]
[762, 0, 0, 0]
[763, 1507, 336, 14170313]
[764, 16, 1, 511]
[765, 0, 0, 0]
[766, 5, 3, 416]
[767, 8, 7, 1553]
[768, 408, 31, 388033]
[769, 0, 0, 0]
[770, 35, 11, 10754]
[771, 98327255, 8571672, 25321305816633841]
[772, 16, 1, 513]
[773, 0, 0, 0]
[774, 0, 0, 0]
[775, 0, 0, 0]
[776, 70, 17, 33511]
[777, 4, 3, 335]
[778, 28, 9, 7073]
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5.References

[1]. A. Bremner and J.W. Jones, On the equation x4+ mx2y2+ y4= z2, J.Number Theory 50 (1995), 286-298.
[2]. Allan J. MacLeod, Elliptic Curves in Recreational Number Theory,arxiv:1610.03430. 
[3]. Allan J. MacLeod, http://web.archive.org/web/20070803035842/http://maths.paisley.ac.uk/allanm/ECRNT/quartic/quartint.htm