1.Introduction

Jim Cullen[1] found small integer solutions of x^4 + y^4 + k = z^2 for k=3,-1,-7. 
We searched the integer solutions of  x^4 + y^4 + k = z^2 for samall k.

2. Claim

1. There is no integer solution of x^4 + y^4 + k = z^2 for k=5,6,10,11,12,13 mod 16.

2. There are infinitely many integer solutions for k=2b^2.
It's obvious by below identity.

(2bma)^4 + (2m^2ba^2)^4 + k^2 = 4b^4(1+2m^4a^4)^2.
a,b,m are arbitrary.

3. There are infinitely many integer solutions for k=-1.
x^4 + y^4 - 1 = z^2.
x^4 + y^4 - 1 = z^2 part2.


3.Numerical solutions

(x,y)<100000

[k  x  y  z]

[2, 1, 1, 2]
[2, 47, 39, 2682]
[2, 149, 53, 22378]
[2, 407, 337, 200842]
[2, 799, 561, 711762]
[2, 1139, 683, 1378642]
[2, 1253, 1101, 1983522]
[2, 11593, 8327, 151230338]
[2, 24289, 12991, 613620002]
[2, 14567, 13601, 281510282]
[2, 68173, 24253, 4684632682]
[2, 72607, 60121, 6391907978]

[3, 3, 2, 10]
[3, 2, 3, 10]
[3, 825, 116, 680758]
[3, 1074, 325, 1158302]
[3, 5877, 646, 34541650]
[3, 8916, 4015, 81113042]
[3, 25617, 5576, 656966830]
[3, 13545, 6296, 187700422]

[4, 2, 2, 6]
[4, 8, 4, 66]
[4, 18, 6, 326]
[4, 32, 8, 1026]
[4, 50, 10, 2502]
[4, 72, 12, 5186]
[4, 98, 14, 9606]
[4, 128, 16, 16386]
[4, 162, 18, 26246]
[4, 200, 20, 40002]
[4, 242, 22, 58566]
[4, 288, 24, 82946]
[4, 338, 26, 114246]
[4, 392, 28, 153666]
[4, 450, 30, 202502]
[4, 512, 32, 262146]
[4, 578, 34, 334086]
[4, 648, 36, 419906]
[4, 722, 38, 521286]
[4, 800, 40, 640002]
[4, 882, 42, 777926]
[4, 968, 44, 937026]
[4, 1058, 46, 1119366]
[4, 1152, 48, 1327106]
[4, 1250, 50, 1562502]
[4, 1352, 52, 1827906]
[4, 1458, 54, 2125766]
[4, 1568, 56, 2458626]
[4, 1682, 58, 2829126]
[4, 1800, 60, 3240002]
[4, 1922, 62, 3694086]
[4, 2048, 64, 4194306]
[4, 2178, 66, 4743686]
[4, 2312, 68, 5345346]
[4, 2450, 70, 6002502]
[4, 2592, 72, 6718466]
[4, 2738, 74, 7496646]
[4, 2888, 76, 8340546]
[4, 3042, 78, 9253766]
[4, 3200, 80, 10240002]
[4, 3362, 82, 11303046]
[4, 3528, 84, 12446786]
[4, 3698, 86, 13675206]
[4, 3872, 88, 14992386]
[4, 4050, 90, 16402502]
[4, 4232, 92, 17909826]
[4, 4418, 94, 19518726]
[4, 4608, 96, 21233666]
[4, 4802, 98, 23059206]
[4, 5000, 100, 25000002]
[4, 5202, 102, 27060806]
[4, 5408, 104, 29246466]
[4, 5618, 106, 31561926]
[4, 5832, 108, 34012226]
[4, 6050, 110, 36602502]
[4, 6272, 112, 39337986]
[4, 6498, 114, 42224006]
[4, 6728, 116, 45265986]
[4, 6962, 118, 48469446]
[4, 7200, 120, 51840002]
[4, 7442, 122, 55383366]
[4, 7688, 124, 59105346]
[4, 7938, 126, 63011846]
[4, 8192, 128, 67108866]
[4, 8450, 130, 71402502]
[4, 8712, 132, 75898946]
[4, 8978, 134, 80604486]
[4, 9248, 136, 85525506]
[4, 9522, 138, 90668486]
[4, 9800, 140, 96040002]
[4, 10082, 142, 101646726]
[4, 10368, 144, 107495426]
[4, 10658, 146, 113592966]
[4, 10952, 148, 119946306]
[4, 11250, 150, 126562502]
[4, 11552, 152, 133448706]
[4, 11858, 154, 140612166]
[4, 12168, 156, 148060226]
[4, 12482, 158, 155800326]
[4, 12800, 160, 163840002]
[4, 13122, 162, 172186886]
[4, 13448, 164, 180848706]
[4, 13778, 166, 189833286]
[4, 14112, 168, 199148546]
[4, 14450, 170, 208802502]
[4, 14792, 172, 218803266]
[4, 15138, 174, 229159046]
[4, 15488, 176, 239878146]
[4, 15842, 178, 250968966]
[4, 16200, 180, 262440002]
[4, 16562, 182, 274299846]
[4, 16928, 184, 286557186]
[4, 17298, 186, 299220806]
[4, 17672, 188, 312299586]
[4, 18050, 190, 325802502]
[4, 18432, 192, 339738626]
[4, 18818, 194, 354117126]
[4, 19208, 196, 368947266]
[4, 19602, 198, 384238406]
[4, 20000, 200, 400000002]
[4, 20402, 202, 416241606]
[4, 20808, 204, 432972866]
[4, 21218, 206, 450203526]
[4, 21632, 208, 467943426]
[4, 22050, 210, 486202502]
[4, 22472, 212, 504990786]
[4, 22898, 214, 524318406]
[4, 23328, 216, 544195586]
[4, 23762, 218, 564632646]
[4, 24200, 220, 585640002]
[4, 24642, 222, 607228166]
[4, 25088, 224, 629407746]
[4, 25538, 226, 652189446]
[4, 25992, 228, 675584066]
[4, 26450, 230, 699602502]
[4, 26912, 232, 724255746]
[4, 27378, 234, 749554886]
[4, 27848, 236, 775511106]
[4, 28322, 238, 802135686]
[4, 28800, 240, 829440002]
[4, 29282, 242, 857435526]
[4, 29768, 244, 886133826]
[4, 30258, 246, 915546566]
[4, 30752, 248, 945685506]
[4, 31250, 250, 976562502]
[4, 31752, 252, 1008189506]
[4, 32258, 254, 1040578566]
[4, 32768, 256, 1073741826]
[4, 33282, 258, 1107691526]
[4, 33800, 260, 1142440002]
[4, 34322, 262, 1177999686]
[4, 34848, 264, 1214383106]
[4, 35378, 266, 1251602886]
[4, 35912, 268, 1289671746]
[4, 36450, 270, 1328602502]
[4, 36992, 272, 1368408066]
[4, 37538, 274, 1409101446]
[4, 38088, 276, 1450695746]
[4, 38642, 278, 1493204166]
[4, 39200, 280, 1536640002]
[4, 39762, 282, 1581016646]
[4, 40328, 284, 1626347586]
[4, 40898, 286, 1672646406]
[4, 41472, 288, 1719926786]
[4, 42050, 290, 1768202502]
[4, 42632, 292, 1817487426]
[4, 43218, 294, 1867795526]
[4, 43808, 296, 1919140866]
[4, 44402, 298, 1971537606]
[4, 45000, 300, 2025000002]
[4, 45602, 302, 2079542406]
[4, 46208, 304, 2135179266]
[4, 46818, 306, 2191925126]
[4, 47432, 308, 2249794626]
[4, 48050, 310, 2308802502]
[4, 48672, 312, 2368963586]
[4, 49298, 314, 2430292806]
[4, 49928, 316, 2492805186]
[4, 50562, 318, 2556515846]
[4, 51200, 320, 2621440002]
[4, 51842, 322, 2687592966]
[4, 52488, 324, 2754990146]
[4, 53138, 326, 2823647046]
[4, 53792, 328, 2893579266]
[4, 54450, 330, 2964802502]
[4, 55112, 332, 3037332546]
[4, 55778, 334, 3111185286]
[4, 56448, 336, 3186376706]
[4, 57122, 338, 3262922886]
[4, 57800, 340, 3340840002]
[4, 58482, 342, 3420144326]
[4, 59168, 344, 3500852226]
[4, 59858, 346, 3582980166]
[4, 60552, 348, 3666544706]
[4, 61250, 350, 3751562502]
[4, 61952, 352, 3838050306]
[4, 62658, 354, 3926024966]
[4, 63368, 356, 4015503426]
[4, 64082, 358, 4106502726]
[4, 64800, 360, 4199040002]
[4, 65522, 362, 4293132486]
[4, 66248, 364, 4388797506]
[4, 66978, 366, 4486052486]
[4, 67712, 368, 4584914946]
[4, 68450, 370, 4685402502]
[4, 69192, 372, 4787532866]
[4, 69938, 374, 4891323846]
[4, 70688, 376, 4996793346]
[4, 71442, 378, 5103959366]
[4, 72200, 380, 5212840002]
[4, 72962, 382, 5323453446]
[4, 73728, 384, 5435817986]
[4, 74498, 386, 5549952006]
[4, 75272, 388, 5665873986]
[4, 76050, 390, 5783602502]
[4, 76832, 392, 5903156226]
[4, 77618, 394, 6024553926]
[4, 78408, 396, 6147814466]
[4, 79202, 398, 6272956806]
[4, 80000, 400, 6400000002]
[4, 80802, 402, 6528963206]
[4, 81608, 404, 6659865666]
[4, 82418, 406, 6792726726]
[4, 83232, 408, 6927565826]
[4, 84050, 410, 7064402502]
[4, 84872, 412, 7203256386]
[4, 85698, 414, 7344147206]
[4, 86528, 416, 7487094786]
[4, 87362, 418, 7632119046]
[4, 88200, 420, 7779240002]
[4, 89042, 422, 7928477766]
[4, 89888, 424, 8079852546]
[4, 90738, 426, 8233384646]
[4, 91592, 428, 8389094466]
[4, 92450, 430, 8547002502]
[4, 93312, 432, 8707129346]
[4, 94178, 434, 8869495686]
[4, 95048, 436, 9034122306]
[4, 95922, 438, 9201030086]
[4, 96800, 440, 9370240002]
[4, 97682, 442, 9541773126]
[4, 98568, 444, 9715650626]
[4, 99458, 446, 9891893766]

[7, 1, 1, 3]
[7, 3, 3, 13]

[8, 2, 1, 5]
[8, 1, 2, 5]
[8, 20, 7, 403]
[8, 11, 10, 157]
[8, 7, 20, 403]
[8, 239, 26, 57125]
[8, 1562, 47, 2439845]
[8, 75, 62, 6813]
[8, 453, 62, 205245]
[8, 13147, 334, 172843645]
[8, 506, 419, 310445]
[8, 712, 421, 537035]
[8, 1299, 548, 1713915]
[8, 3124, 689, 9770915]
[8, 1070, 763, 1284413]
[8, 2215, 788, 4945363]
[8, 5604, 809, 31411635]
[8, 9181, 1600, 84329627]
[8, 24029, 2316, 577417755]
[8, 8735, 2462, 76540613]
[8, 689, 3124, 9770915]
[8, 9172, 3263, 84796675]
[8, 4402, 3645, 23494893]
[8, 10568, 5123, 114724955]
[8, 22467, 8200, 509224923]
[8, 13427, 11118, 218590605]
[8, 19391, 14054, 424730885]
[8, 28783, 15930, 866455077]
[8, 34299, 24598, 1322900925]
[8, 66338, 48771, 5002420845]

[9, 6, 6, 51]
[9, 6090, 2310, 37470003]
[9, 57408, 7318, 3296113541]

[14, 1, 1, 4]
[14, 13, 7, 176]
[14, 69, 13, 4764]
[14, 41, 35, 2080]
[14, 47, 47, 3124]
[14, 791, 605, 724880]
[14, 2807, 2179, 9199264]

[15, 6, 5, 44]
[15, 25, 18, 704]
[15, 4905, 1282, 24115096]
[15, 7983, 4525, 66936881]
[15, 71445, 52943, 5823346879]

[-2, 33, 15, 1112]
[-2, 427, 115, 182808]
[-2, 295, 121, 88248]
[-2, 291, 261, 108680]
[-2, 121, 295, 88248]
[-2, 8577, 705, 73566608]
[-2, 15279, 12465, 280427648]
[-2, 34147, 16325, 1196086248]
[-2, 35653, 22115, 1361976648]
[-2, 51231, 23247, 2679675880]

[-7, 2, 2, 5]
[-7, 80, 14, 6403]
[-7, 478, 26, 228485]
[-7, 1604, 132, 2572875]
[-7, 794, 700, 798467]
[-7, 20776, 2746, 431708035]
[-7, 5566, 3640, 33694723]
[-7, 68104, 6700, 4638372043]
[-7, 34720, 8198, 1207350397]
[-7, 29474, 12778, 883927675]

[-8, 2, 1, 3]
[-8, 6, 3, 37]
[-8, 23, 22, 717]
[-8, 46, 29, 2277]
[-8, 74, 71, 7443]
[-8, 813, 78, 660997]
[-8, 357, 162, 130123]
[-8, 3507, 162, 12299077]
[-8, 966, 279, 936397]
[-8, 1103, 1058, 1653213]
[-8, 3554, 3409, 17163747]
[-8, 9474, 4299, 91639637]
[-8, 9774, 4983, 98705267]
[-8, 17583, 16566, 413393507]
[-8, 24202, 20131, 712264677]
[-8, 97662, 26019, 9561862043]
[-8, 47341, 39166, 2715865467]
[-8, 87909, 47958, 8062984907]
[-8, 52967, 50806, 3812308653]
[-8, 26019, 97662, 9561862043]

[-13, 2, 1, 2]
[-13, 4, 3, 18]
[-13, 11, 4, 122]
[-13, 14, 7, 202]
[-13, 13, 12, 222]
[-13, 28, 13, 802]
[-13, 181, 16, 32762]
[-13, 108, 67, 12498]
[-13, 122, 87, 16698]
[-13, 448, 133, 201482]
[-13, 709, 148, 503158]
[-13, 871, 214, 760022]
[-13, 1642, 353, 2699042]
[-13, 796, 701, 801838]
[-13, 10394, 817, 108037298]
[-13, 3062, 1753, 9866602]
[-13, 2633, 1834, 7705562]
[-13, 14134, 8241, 210998478]
[-13, 38864, 9331, 1512917918]
[-13, 66137, 20298, 4393464042]

[-15, 9094, 6000, 90196609]
[-15, 39900, 8114, 1593370751]

4.Reference

[1]:Jim Cullen, x^4 + y^4 - 1 = z^2