1.Introduction

By Tito Piezas[1], it seems that a^4 + b^4 = c^4 + 12d^4 has a follwing parametric solution.
  
(192n^7)^4 + (192n^8-24n^4-1)^4 = (192n^8+24n^4-1)^4 + 12(2n)^4.(Noam Elkies)

And a^4 + b^4 + 4c^4 = d^4 has a parametric solution as
(p^4-2q^4)^4 + (2p^3q)^4 + 4(2pq^3)^4 = (p^4+2q^4)^4.(Carmichael)

I found some solutions before as follows.(A...F)

A: (12x^8 + 6x^4 - 1)^4 + 3(2x)^4    = (12x^8 - 6x^4 - 1)^4 + 4(12x^7)^4.

B: (48x^8 + 12x^4 - 1)^4 + 6(2x)^4   = (48x^8 - 12x^4 - 1)^4 + 2(48x^7)^4.

C: (3x^8 + 3x^4 - 1)^4 + 24(x)^4     = (3x^8 - 3x^4 - 1)^4 + 8(3x^7)^4.

D: (972x^8 + 54x^4 - 1)^4 + 27(2x)^4 = (972x^8 - 54x^4 - 1)^4 + 36(324x^7)^4.

E: (24x^4 + 1)^4 = (24x^4 - 1)^4 + 27(8x^3)^4 + 12(2x)^4.

F: (18x^4 + 1)^4 = (18x^4 - 1)^4 + 36(6x^3)^4 + 9(2x)^4.

I searched numerical solutions this time. 

2. Search results

Numeric solutions of x^4 + ay^4 = z^4 + bt^4 where (a,b) < 10.
Max(x,y,z,t)<100000


   a, b         x       y       z        t
[  2  1] [     143     180     224      15]
[  2  1] [     345     452     272     551]
[  2  1] [     680     969    1183     369]
[  2  1] [    5375     964    2976    5247]
[  3  1] [      ?        ?      ?       ? ]
[  3  2] [    1560     357    1243    1157]
[  3  2] [    4680     529    4271    2929]
[  4  1] [     897    1660     129    2360]
[  4  1] [    2093    1420    2440     301]
[  4  2] [ 1169407 1157520 1484801 1203120] Andreas Ensenhans and Jorg Jahnel[2]
[  4  3] [    infinite: identity A        ]
[  5  1] [       1       2       3       0]
[  5  1] [      51      14      29      50]
[  5  2] [      19      14       7      20]
[  5  2] [      81     276     401     200]
[  5  3] [      ?        ?      ?       ? ]
[  5  4] [      17     258     307     240]
[  5  4] [      27      14      17      20]
[  6  1] [       1       2       2       3]
[  6  2] [       infinite: identity B     ]
[  6  3] [    2867    5660    2419    6740]
[  6  4] [      ?        ?      ?       ? ]
[  6  5] [       3       0       1       2]
[  6  5] [      15      14      23       4]
[  6  5] [      25      28      13      30]
[  6  5] [      45      34      59       4]
[  6  5] [     196      75      47     135]
[  6  5] [    3480    2633    4567      23]
[  6  5] [    4612    2355    4379    2709]
[  6  5] [    6269    1220    6211    1932]
[  7  1] [       5      16      26       7]
[  7  1] [      25     284     409     364]
[  7  1] [     263      20     259     130]
[  7  2] [      11       2       9       8]
[  7  2] [     447     104     449      84]
[  7  3] [    1059    2506    3665    2384]
[  7  4] [      ?        ?      ?       ? ]
[  7  5] [     763     740     453     832]
[  7  6] [     987    2810    2759    2820]
[  7  6] [    1330     723     647     949]
[  7  6] [    2625    3418    5627     484]
[  7  6] [    7189    4860    7811    4670]
[  8  1] [   88769   10350   88800    8417]
[  8  2] [       5      22      37       4]
[  8  2] [      15      94     111     124]
[  8  2] [     405     252     277     404]
[  8  2] [     655      92     631     338]
[  8  2] [    1215     236     831     964]
[  8  3] [      29       4       9      22]
[  8  3] [      47      92     147      78]
[  8  3] [      73      16      69      38]
[  8  3] [     407     112     411      86]
[  8  4] [      33      38      65       8]
[  8  4] [      73     348     585      88]
[  8  5] [    9121   11760    4001   13368]
[  8  6] [      ?        ?      ?       ? ]
[  8  7] [     330     197     131     241]
[  9  1] [     133     596     965     720]
[  9  2] [      ?        ?      ?       ? ]
[  9  3] [       5       2       1       4]
[  9  3] [      17      46      35      60]
[  9  3] [     805     316     613     592]
[  9  3] [    3335     856    2809    2168]
[  9  4] [      89      62     117      38]
[  9  4] [     173      18      57     122]
[  9  4] [     287      68     289      24]
[  9  4] [     475    1388    2405      72]
[  9  4] [     523     346     351     466]
[  9  4] [    1061     614    1263     146]
[  9  4] [    3137    2098    4029    1178]
[  9  4] [    5927    6094    1419    7642]
[  9  4] [    6209     908    5761    3144]
[  9  4] [    7309     606    2535    5150]
[  9  5] [       1      10      17       6]
[  9  6] [     137      60     145      46]
[  9  7] [      ?        ?      ?       ? ]
[  9  8] [     600     257     641      47]



3.References

[1].Tito Piezas:http://sites.google.com/site/tpiezas/updates03

[2].Andreas Ensenhans and Jorg Jahnel: The Diophantine Equation x^4+2y^4=z^4+4w^4




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