diophantine equation x^6 + ky^3 = z^6 + kw^3 dioph459e

                 x^6 + ky^3 = z^6 + kw^3 

A.S. Janfada et al.[1] proved that x^6 + ky^3 = z^6 + kw^3 has infinitely many rational solutions with k <500.
Since we can't apply the previous result, we searched the Mordell-Weil rank with 500 < k <1000.
We show the diophantine equation x^6 + ky^3 = z^6 + kw^3 has infinitely many rational solutions with 500 < k <1000.
There always seems to be a solution for any k.

When k=p^3±q^3, there is a obvious soution p^6 + k(±q)^3 = q^6 + kp^3.


x^6 + ky^3 = z^6 + kw^3..........................................................(1)

Substitute  x=p, z=q, y=t+s, w=-t+s to (1), then we obtain

p^6+2kt^3+6kts^2-q^6=0

Multiplying both side of above equation by 6kt, we obtain

v^2 = -12k^2t^4-6k(p^6-q^6)t.....................................................(2)
v = 6kts
 
This quartic curve (2) can be transformed to the elliptic curve (3).

Y^2 = X^3-432k^4(p^6-q^6)^2......................................................(3)

t = -6k(p^6-q^6)/X
s = 1/6Y/(kX)

We searched the rank of elliptic curve (3) using Online Magma Calculator.

Since the rank of elliptic curve (3) is positive, equation (1) has infinitely many rational solutions with 500 < k <1000.


Search result.
500 < k <1000

[k] [rank] [p,q]

[500] [2] [2, 1],[625] [1] [4, 1],[750] [1] [2, 1],[875] [1] [2, 1]
[501] [1] [2, 1],[626] [2] [2, 1],[751] [1] [2, 1],[876] [1] [2, 1]
[502] [1] [2, 1],[627] [1] [2, 1],[752] [1] [2, 1],[877] [1] [2, 1]
[503] [2] [2, 1],[628] [1] [4, 1],[753] [1] [2, 1],[878] [2] [3, 1]
[504] [2] [4, 1],[629] [1] [3, 1],[754] [1] [2, 1],[879] [1] [2, 1]
[505] [1] [2, 1],[630] [2] [3, 1],[755] [1] [3, 1],[880] [1] [4, 1]
[506] [1] [4, 1],[631] [1] [2, 1],[756] [1] [3, 1],[881] [1] [3, 1]
[507] [2] [2, 1],[632] [1] [2, 1],[757] [1] [2, 1],[882] [1] [3, 1]
[508] [1] [4, 1],[633] [1] [3, 1],[758] [1] [4, 1],[883] [1] [2, 1]
[509] [1] [4, 1],[634] [1] [2, 1],[759] [1] [3, 1],[884] [1] [2, 1]
[510] [1] [2, 1],[635] [2] [2, 1],[760] [1] [4, 1],[885] [1] [4, 1]
[511] [1] [2, 1],[636] [2] [2, 1],[761] [1] [2, 3],[886] [1] [2, 1]
[512] [1] [2, 1],[637] [1] [2, 1],[762] [1] [2, 1],[887] [1] [4, 1]
[513] [1] [2, 1],[638] [1] [3, 1],[763] [1] [2, 1],[888] [2] [2, 1]
[514] [1] [2, 1],[639] [1] [3, 1],[764] [1] [2, 1],[889] [1] [2, 1]
[515] [1] [4, 1],[640] [1] [2, 1],[765] [1] [2, 1],[890] [1] [1, 3]
[516] [1] [2, 1],[641] [1] [7, 1],[766] [1] [2, 1],[891] [1] [2, 1]
[517] [1] [2, 1],[642] [1] [3, 1],[767] [1] [4, 1],[892] [2] [2, 1]
[518] [2] [3, 1],[643] [1] [2, 1],[768] [1] [2, 1],[893] [1] [4, 1]
[519] [1] [2, 1],[644] [1] [2, 1],[769] [1] [2, 1],[894] [1] [3, 1]
[520] [2] [2, 1],[645] [1] [2, 1],[770] [1] [4, 1],[895] [1] [2, 1]
[521] [1] [3, 1],[646] [1] [2, 1],[771] [1] [2, 1],[896] [1] [4, 1]
[522] [1] [2, 1],[647] [1] [3, 1],[772] [1] [4, 1],[897] [1] [2, 1]
[523] [1] [2, 1],[648] [1] [3, 1],[773] [1] [3, 1],[898] [1] [2, 1]
[524] [1] [2, 1],[649] [1] [2, 1],[774] [1] [3, 1],[899] [1] [3, 1]
[525] [1] [2, 1],[650] [1] [2, 1],[775] [2] [2, 1],[900] [1] [2, 1]
[526] [1] [2, 1],[651] [1] [3, 1],[776] [1] [2, 1],[901] [1] [2, 1]
[527] [1] [1, 4],[652] [2] [2, 1],[777] [1] [3, 1],[902] [1] [4, 1]
[528] [1] [3, 1],[653] [1] [4, 1],[778] [1] [2, 1],[903] [1] [3, 1]
[529] [2] [3, 1],[654] [1] [2, 1],[779] [2] [5, 1],[904] [2] [3, 1]
[530] [1] [4, 1],[655] [1] [2, 1],[780] [2] [2, 1],[905] [1] [4, 1]
[531] [1] [3, 1],[656] [1] [2, 1],[781] [1] [2, 1],[906] [1] [2, 1]
[532] [2] [2, 1],[657] [1] [2, 1],[782] [1] [3, 1],[907] [1] [2, 1]
[533] [1] [4, 1],[658] [1] [2, 1],[783] [2] [3, 1],[908] [1] [2, 1]
[534] [1] [4, 1],[659] [1] [8, 1],[784] [1] [3, 1],[909] [1] [2, 1]
[535] [1] [2, 1],[660] [1] [2, 1],[785] [1] [4, 1],[910] [1] [2, 1]
[536] [1] [2, 1],[661] [1] [2, 1],[786] [1] [4, 1],[911] [1] [4, 1]
[537] [1] [2, 1],[662] [1] [7, 1],[787] [1] [2, 1],[912] [1] [2, 1]
[538] [1] [2, 1],[663] [1] [2, 1],[788] [1] [2, 1],[913] [1] [2, 1]
[539] [2] [2, 1],[664] [1] [4, 1],[789] [1] [2, 1],[914] [3] [4, 1]
[540] [1] [2, 1],[665] [1] [3, 1],[790] [1] [2, 1],[915] [1] [2, 1]
[541] [1] [2, 1],[666] [1] [2, 1],[791] [1] [3, 1],[916] [2] [3, 1]
[542] [1] [4, 1],[667] [1] [2, 1],[792] [1] [2, 1],[917] [1] [3, 1]
[543] [1] [3, 1],[668] [1] [2, 1],[793] [1] [2, 1],[918] [1] [2, 1]
[544] [1] [2, 1],[669] [1] [3, 1],[794] [1] [1, 4],[919] [1] [2, 1]
[545] [1] [4, 1],[670] [1] [2, 1],[795] [1] [3, 1],[920] [1] [2, 1]
[546] [1] [2, 1],[671] [1] [4, 1],[796] [1] [4, 1],[921] [1] [3, 1]
[547] [1] [2, 1],[672] [1] [2, 1],[797] [2] [3, 1],[922] [1] [2, 1]
[548] [1] [2, 1],[673] [1] [2, 1],[798] [1] [2, 1],[923] [1] [4, 1]
[549] [1] [5, 1],[674] [1] [3, 1],[799] [1] [2, 1],[924] [1] [3, 1]
[550] [1] [3, 1],[676] [1] [3, 1],[800] [1] [2, 1],[925] [2] [3, 1]
[551] [1] [4, 1],[676] [1] [3, 1],[801] [1] [3, 1],[926] [1] [3, 1]
[552] [1] [2, 1],[677] [1] [4, 1],[802] [1] [2, 1],[927] [1] [2, 1]
[553] [1] [2, 1],[678] [2] [2, 1],[803] [1] [4, 1],[928] [1] [2, 1]
[554] [1] [4, 1],[679] [1] [2, 1],[804] [1] [2, 1],[929] [1] [8, 1]
[555] [1] [2, 1],[680] [1] [2, 1],[805] [1] [2, 1],[930] [1] [3, 1]
[556] [2] [3, 1],[681] [1] [2, 1],[806] [1] [4, 1],[931] [1] [2, 1]
[557] [1] [3, 1],[682] [1] [2, 1],[807] [1] [2, 1],[932] [1] [2, 1]
[558] [1] [3, 1],[683] [1] [3, 1],[808] [1] [4, 1],[933] [1] [2, 1]
[559] [1] [2, 1],[684] [2] [2, 1],[809] [1] [3, 1],[934] [1] [5, 2]
[560] [1] [2, 1],[685] [1] [2, 1],[810] [2] [2, 1],[935] [1] [3, 1]
[561] [2] [2, 1],[686] [1] [4, 1],[811] [1] [2, 1],[936] [1] [2, 1]
[562] [1] [2, 1],[687] [1] [3, 1],[812] [1] [2, 1],[937] [1] [2, 1]
[563] [2] [2, 1],[688] [2] [3, 1],[813] [1] [3, 1],[938] [2] [2, 1]
[564] [2] [2, 1],[689] [1] [4, 1],[814] [1] [2, 1],[939] [1] [3, 1]
[565] [1] [2, 1],[690] [1] [2, 1],[815] [2] [3, 1],[940] [2] [3, 1]
[566] [1] [3, 1],[691] [1] [2, 1],[816] [1] [3, 1],[941] [1] [3, 2]
[567] [1] [3, 1],[692] [1] [2, 1],[817] [1] [2, 1],[942] [1] [2, 1]
[568] [2] [2, 1],[693] [1] [3, 1],[818] [1] [3, 1],[943] [1] [2, 1]
[569] [1] [8, 1],[694] [1] [2, 1],[819] [1] [2, 1],[944] [1] [2, 1]
[570] [1] [3, 1],[695] [1] [4, 1],[820] [1] [3, 1],[945] [1] [3, 1]
[571] [1] [2, 1],[696] [1] [2, 1],[821] [1] [6, 1],[946] [1] [2, 1]
[572] [1] [2, 1],[697] [1] [2, 1],[822] [1] [7, 1],[947] [1] [1, 4]
[573] [1] [2, 1],[698] [1] [7, 1],[823] [1] [2, 1],[948] [1] [2, 1]
[574] [1] [2, 1],[699] [1] [2, 1],[824] [1] [2, 1],[949] [1] [2, 1]
[575] [1] [2, 1],[700] [1] [2, 1],[825] [1] [3, 1],[950] [1] [2, 1]
[576] [1] [2, 1],[701] [1] [4, 1],[826] [1] [2, 1],[951] [1] [2, 1]
[577] [1] [2, 1],[702] [1] [3, 1],[827] [1] [3, 1],[952] [1] [4, 1]
[578] [1] [2, 1],[703] [1] [2, 1],[828] [2] [3, 1],[953] [1] [1, 3]
[579] [1] [3, 1],[704] [1] [4, 1],[829] [1] [2, 1],[954] [1] [1, 5]
[580] [1] [4, 1],[705] [1] [3, 1],[830] [1] [4, 1],[955] [1] [2, 1]
[581] [2] [3, 1],[706] [1] [2, 1],[831] [1] [3, 1],[956] [1] [2, 1]
[582] [1] [2, 1],[707] [1] [4, 1],[832] [1] [2, 1],[957] [1] [3, 1]
[583] [1] [2, 1],[708] [1] [3, 1],[833] [1] [4, 1],[958] [1] [2, 1]
[584] [1] [2, 1],[709] [1] [2, 1],[834] [1] [2, 1],[959] [1] [4, 1]
[585] [1] [2, 1],[710] [1] [4, 1],[835] [1] [2, 1],[960] [1] [2, 1]
[586] [1] [2, 1],[711] [2] [4, 1],[836] [1] [2, 1],[961] [1] [2, 1]
[587] [2] [3, 1],[712] [1] [3, 1],[837] [1] [2, 1],[962] [1] [3, 1]
[588] [1] [2, 1],[713] [1] [1, 4],[838] [1] [2, 1],[963] [1] [3, 1]
[589] [1] [2, 1],[714] [1] [3, 1],[839] [1] [7, 1],[964] [1] [3, 1]
[590] [1] [4, 1],[715] [1] [2, 1],[840] [1] [2, 1],[965] [2] [2, 1]
[591] [1] [2, 1],[716] [1] [2, 1],[841] [2] [3, 1],[966] [2] [2, 1]
[592] [1] [4, 1],[717] [1] [2, 1],[842] [2] [5, 2],[967] [1] [3, 1]
[593] [1] [3, 1],[718] [1] [2, 1],[843] [1] [2, 1],[968] [2] [3, 1]
[594] [1] [2, 1],[719] [1] [4, 1],[844] [1] [6, 1],[969] [1] [2, 1]
[595] [1] [2, 1],[720] [1] [2, 1],[845] [1] [3, 1],[970] [1] [3, 1]
[596] [1] [2, 1],[721] [1] [2, 1],[846] [1] [2, 1],[971] [1] [3, 1]
[597] [1] [3, 1],[722] [2] [2, 1],[847] [1] [3, 1],[972] [1] [3, 1]
[598] [1] [2, 1],[723] [1] [3, 1],[848] [1] [2, 1],[973] [1] [2, 1]
[599] [1] [4, 1],[724] [2] [2, 1],[849] [1] [3, 1],[974] [2] [2, 1]
[600] [1] [2, 1],[725] [1] [2, 1],[850] [2] [2, 1],[975] [1] [2, 1]
[601] [1] [2, 1],[726] [2] [2, 1],[851] [1] [4, 1],[976] [1] [4, 1]
[602] [1] [3, 1],[727] [1] [2, 1],[852] [1] [3, 1],[977] [1] [1, 4]
[603] [1] [2, 1],[728] [1] [2, 1],[853] [1] [2, 1],[978] [1] [2, 1]
[604] [1] [3, 1],[729] [1] [2, 1],[854] [1] [3, 1],[979] [1] [2, 1]
[605] [1] [2, 1],[730] [1] [2, 1],[855] [1] [3, 1],[980] [1] [2, 1]
[606] [1] [3, 1],[731] [1] [4, 1],[856] [1] [3, 1],[981] [1] [2, 1]
[607] [1] [2, 1],[732] [1] [2, 1],[857] [1] [4, 1],[982] [1] [2, 1]
[608] [1] [4, 1],[733] [1] [2, 1],[858] [1] [3, 1],[983] [1] [1, 4]
[609] [1] [2, 1],[734] [1] [7, 1],[859] [1] [2, 1],[984] [1] [2, 1]
[610] [1] [2, 1],[735] [1] [2, 1],[860] [1] [2, 1],[985] [1] [3, 1]
[611] [1] [3, 1],[736] [1] [2, 1],[861] [1] [2, 1],[986] [1] [4, 1]
[612] [1] [2, 1],[737] [1] [3, 1],[862] [1] [2, 1],[987] [1] [2, 1]
[613] [1] [2, 1],[738] [1] [2, 1],[863] [1] [3, 1],[988] [2] [3, 1]
[614] [2] [3, 1],[739] [1] [2, 1],[864] [2] [2, 1],[989] [1] [3, 1]
[615] [1] [3, 1],[740] [1] [2, 1],[865] [1] [2, 1],[990] [1] [2, 1]
[616] [1] [4, 1],[741] [1] [3, 1],[866] [2] [5, 1],[991] [1] [2, 1]
[617] [1] [4, 1],[742] [1] [2, 1],[867] [1] [2, 1],[992] [2] [2, 1]
[618] [1] [2, 1],[743] [2] [3, 1],[868] [1] [4, 1],[993] [1] [3, 1]
[619] [1] [2, 1],[744] [1] [3, 1],[869] [1] [4, 1],[994] [1] [2, 1]
[620] [1] [2, 1],[745] [1] [2, 1],[870] [1] [2, 1],[995] [1] [4, 1]
[621] [1] [4, 1],[746] [1] [3, 1],[871] [1] [2, 1],[996] [1] [3, 1]
[622] [1] [2, 1],[747] [1] [2, 1],[872] [1] [2, 1],[997] [1] [2, 1]
[623] [1] [4, 1],[748] [1] [3, 1],[873] [1] [5, 1],[998] [1] [3, 1]
[624] [1] [2, 1],[749] [2] [2, 1],[874] [1] [2, 1],[999] [1] [2, 1]



4.Reference

[1]:  H. Shabani-Solt, N. Yousefnejad, and A. S. Janfada, On theDiophantine equation x^6 + ky^3 = z^6 + kw^3 , Iran. J. Math. Sci.Inform., Vol. 15, No. 1 (2020)
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