1.Introduction

We treat Generalised Taxicab Numbers problem, Taxicab(4,3,m).
According to Duncan Moore's[1] website Generalised Taxicab Numbers and Cabtaxi Numbers, almost smallest solutions have the form 
such as a^4+b^4+(a+b)^4.
Solutions for m=22,26,29,31,33,34,35,37,38,39 were not found with m<40.
Why does the form a^4+b^4+(a+b)^4 give many smallest solutions?
a^4+b^4+(a+b)^4=2(a^2+ab+b^2)^2 is known as Proth's identity.
Anyway, we searched the unknown solutions using Proth's identity.

2.Method

a^4+b^4+(a+b)^4=N.
We use Proth's identity below.
a^4+b^4+(a+b)^4=2(a^2+ab+b^2)^2.
Hence let consider the all integral solutions of a^2+ab+b^2=n.

We assume n doesn't have large prime factos, so let n=7^(n1)13^(n2)19^(n3)31^(n4)37^(n5)43^(n6)61^(n7)67^(n8)73^(n9)79^(n10).

Let play with Eisenstein prime integer, then above prime factors are factoed as (a+bw)(a+bw^2) below.

7=(3+w)(3+w^2),13=(4+w)(4+w^2),19=(5+2w)(5+2w^2),31=(6+w)(6+w^2),37=(7+3w)(7+3w^2)
43=(7+6w)(7+6w^2),61=(9+4w)(9+4w^2),67=(9+7w)(9+7w^2),73=(9+8w)(9+8w^2),79=(10+7w)(10+7w^2)

Constructing all conbination of above prime factors, we can obtain all solutions of a^2+ab+b^2=n.

Example: m=8, a^2+ab+b^2=7*13*19*31

(3+w)(4+w)(5+2w)(6+w)= 218+243w, [a,b,a+b]=[243, 218, 25]
(3+w)(4+w)(5+2w)(6+w^2)= 255+197w, [a,b,a+b]=[197,255, 58]
(3+w)(4+w)(5+2w^2)(6+w)= 262+85w, [a,b,a+b]=[85, 262, 177]
(3+w)(4+w)(5+2w^2)(6+w^2)= 233+3w, [a,b,a+b]=[3, 233, 230]
(3+w)(4+w^2)(5+2w)(6+w)= 265+163w, [a,b,a+b]=[163, 265, 102]
(3+w)(4+w^2)(5+2w)(6+w^2)= 263+90w, [a,b,a+b]=[90, 263, 173]
(3+w)(4+w^2)(5+2w^2)(6+w)= 207-43w, [a,b,a+b]=[43, 207, 250]
(3+w)(4+w^2)(5+2w^2)(6+w^2)= 145-122w, [a,b,a+b]=[122, 145, 267]

3.Results

Solutions are smallest in search range, but not always smallest in general.

m=22:   43771   43771       0 =    44200   43329     871 =    45201   42185    3016 =    45584   41699    3885 =
        45815   41391    4424 =    46176   40885    5291 =    46845   39856    6989 =    47320   39039    8281 =
        47619   38480    9139 =    48165   37349   10816 =    48545   36456   12089 =    48919   35464   13455 =
        49200   34621   14579 =    49321   34225   15096 =    49504   33579   15925 =    49751   32591   17160 =
        50024   31265   18759 =    50141   30576   19565 =    50299   29440   20859 =    50464   27669   22795 =
        50505   26936   23569 =    50531   26195   24336
        
m=26:   57792   57491     301 =    59017   56161    2856 =    59899   55083    4816 =    60333   54509    5824 =
        61208   53249    7959 =    62608   50869   11739 =    62944   50211   12733 =    63167   49751   13416 =
        63609   48776   14833 =    64267   47131   17136 =    64421   46704   17717 =    64792   45591   19201 =
        65016   44849   20167 =    65311   43767   21544 =    65408   43379   22029 =    65661   42269   23392 =
        65819   41483   24336 =    65833   41409   24424 =    65919   40936   24983 =    66031   40264   25767 =
        66104   39783   26321 =    66339   37856   28483 =    66456   36439   30017 =    66479   36071   30408 =
        66521   35217   31304 =    66541   34624   31917
        
m=29:   43679   43152     527 =    44296   42483    1813 =    44797   41893    2904 =    45248   41321    3927 =   
        45557   40904    4653 =    45787   40579    5208 =    46221   39928    6293 =    46704   39137    7567 =
        46872   38843    8029 =    47067   38488    8579 =    47313   38017    9296 =    47432   37779    9653 =
        47663   37296   10367 =    47833   36921   10912 =    47971   36603   11368 =    48277   35848   12429 =
        48329   35712   12617 =    48608   34937   13671 =    48907   34003   14904 =    49128   33221   15907 =
        49329   32417   16912 =    49693   30597   19096 =    49847   29568   20279 =    49952   28679   21273 =
        50029   27832   22197 =    50071   27223   22848 =    50096   26753   23343 =    50127   25823   24304 
        50133   25421   24712
        
m=31:   45477   45322     155 =    45983   44793    1190 =    46410   44317    2093 =    47005   43603    3402 =
        47213   43338    3875 =    47398   43095    4303 =    47585   42842    4743 =    47782   42567    5215 =
        47957   42315    5642 =    48650   41237    7413 =    48707   41142    7565 =    49135   40393    8742 =
        49283   40118    9165 =    49777   39130   10647 =    49910   38843   11067 =    50305   37927   12378 =
        50427   37622   12805 =    50778   36673   14105 =    50883   36365   14518 =    50995   36022   14973 =
        51290   35033   16257 =    51702   33355   18347 =    51863   32550   19313 =    51930   32177   19753 =
        52042   31485   20557 =    52143   30758   21385 =    52262   29687   22575 =    52297   29295   23002 =
        52325   28938   23387 =    52390   27807   24583 =    52423   26278     261
        
m=33:   57928   56121    1807 =    58253   55757    2496 =    59293   54496    4797 =    59423   54327    5096 =
        59696   53963    5733 =    59904   53677    6227 =    61152   51779    9373 =    61971   50323   11648 =
        62283   49712   12571 =    62504   49257   13247 =    62853   48496   14357 =    62973   48221   14752 =
        63193   47697   15496 =    63379   47232   16147 =    63504   46907   16597 =    63583   46696   16887 =
        64129   45097   19032 =    64272   44629   19643 =    64337   44408   19929 =    64571   43563   21008 =
        64792   42679   22113 =    64987   41808   23179 =    65096   41273   23823 =    65233   40537   24696 =
        65352   39823   25529 =    65429   39312   26117 =    65679   37192   28487 =    65728   36621   29107 =
        65767   36088   29679 =    65793   35672   30121 =    65832   34879   30953 =    65867   33563   32304 =
        65871   32968   32903
        
m=34:   44506   44163     343 =    44733   43927     806 =    44869   43782    1087 =    45217   43398    1819 = 
        45357   43238    2119 =    45818   42687    3131 =    46319   42042    4277 =    46438   41881    4557 =
        46631   41613    5018 =    47593   40131    7462 =    47793   39787    8006 =    47957   39494    8463 =
        48347   38753    9594 =    48763   37882   10881 =    48902   37569   11333 =    49062   37193   11869 =
        49306   36583   12723 =    49498   36067   13431 =    49569   35867   13702 =    49817   35123   14694 =
        49881   34918   14963 =    50127   34069   16058 =    50219   33722   16497 =    50323   33306   17017 =
        50406   32953   17453 =    50619   31937   18682 =    50717   31398   19319 =    50778   31031   19747 =
        50897   30219   20678 =    51103   28197   22906 =    51119   27962   23157 =    51142   27573   23569 =
        51181   26598   24583 =    51194   25727   25467

m=35:   59591   58737     854 =    60326   57939    2387 =    60949   57211    3738 =    61411   56637    4774 =
        62363   55349    7014 =    62762   54761    8001 =    63021   54362    8659 =    63301   53914    9387 =
        64449   51863   12586 =    64666   51429   13237 =    65437   49731   15706 =    65639   49238   16401 =
        65813   48794   17019 =    65839   48726   17113 =    66013   48259   17754 =    66262   47551   18711 =
        66381   47194   19187 =    66609   46471   20138 =    66619   46438   20181 =    66774   45911   20863 =
        66906   45437   21469 =    67263   44009   23254 =    67487   42966   24521 =    67494   42931   24563 =
        67641   42154   25487 =    67738   41589   26149 =    67921   40362   27559 =    67987   39846   28141 =
        68054   39263   28791 =    68138   38409   29729 =    68231   37169   31062 =    68262   36611   31651 =
        68266   36529   31737 =    68299   35686   32613 =    68313   35119   33194

m=37:   52455   51974     481 =    52741   51675    1066 =    53365   50986    2379 =    54314   49829    4485 =
        54585   49471    5114 =    55146   48685    6461 =    55279   48489    6790 =    55510   48139    7371 =
        55965   47411    8554 =    56966   45591   11375 =    56979   45565   11414 =    57395   44694   12701 =
        57669   44074   13595 =    57850   43641   14209 =    58006   43251   14755 =    58175   42809   15366 =
        58326   42395   15931 =    58409   42159   16250 =    58645   41451   17194 =    58721   41210   17511 =
        58730   41181   17549 =    58851   40781   18070 =    58985   40314   18671 =    59099   39894   19205 =
        59514   38129   21385 =    59605   37674   21931 =    59654   37415   22239 =    59910   35839   24071 =
        59969   35399   24570 =    60081   34426   25655 =    60111   34121   25990 =    60151   33670   26481 =
        60199   33030   27169 =    60229   32539   27690 =    60255   32006   28249 =    60281   31226   29055 =
        60294   30199   30095

m=38:   53501   53438      63 =    53963   52962    1001 =    55657   50974    4683 =    55874   50687    5187 = 
        55993   50526    5467 =    56433   49907    6526 =    56642   49599    7043 =    57057   48958    8099 =
        57342   48493    8849 =    57827   47649   10178 =    58149   47047   11102 =    58317   46718   11599 =
        58363   46626   11737 =    58498   46351   12147 =    58526   46293   12233 =    58818   45667   13151 =
        59213   44751   14462 =    59423   44226   15197 =    59703   43477   16226 =    60002   42603   17399 =
        60151   42133   18018 =    60263   41762   18501 =    60382   41349   19033 =    60851   39473   21378 =
        60943   39042   21901 =    60957   38974   21983 =    61022   38649   22373 =    61117   38143   22974 =
        61243   37401   23842 =    61278   37177   24101 =    61399   36322   25077 =    61474   35707   25767 =
        61601   34403   27198 =    61649   33747   27902 =    61677   33278   28399 =    61698   32851   28847 =
        61726   32053   29673 =    61737   31499   30238

m=39:   50673   50540     133 =    50953   50253     700 =    51660   49483    2177 =    51943   49155    2788 = 
        52465   48517    3948 =    53333   47348    5985 =    53580   46987    6593 =    53668   46855    6813 =
        53907   46487    7420 =    54363   45743    8620 =    54460   45577    8883 =    54663   45220    9443 =
        54817   44940    9877 =    55593   43388   12205 =    55727   43092   12635 =    55747   43047   12700 =
        55895   42707   13188 =    55972   42525   13447 =    56308   41685   14623 =    56525   41097   15428 =
        56635   40783   15852 =    56677   40660   16017 =    56772   40375   16397 =    57372   38297   19075 =
        57393   38213   19180 =    57485   37833   19652 =    57855   36043   21812 =    57932   35595   22337 =
        57988   35245   22743 =    58085   34577   23508 =    58107   34412   23695 =    58205   33593   24612 =
        58228   33375   24853 =    58268   32963   25305 =    58273   32908   25365 =    58387   31255   27132 =
        58415   30548   27867 =    58425   30172   28253 =    58433   29685   28748
        
Only N is shown for m>=40.

m=40: N =    925411853648116562       
m=41: N =  16506566432427091202       
m=42: N =  11008199896727970482       
m=43: N = 118344783539056789442       
m=44: N =   9208303883062680578       
m=45: N = 122972088779971620098       
m=46: N = 155973999414069951842       
m=48: N =    712655775674871938       
m=54: N =  23514557175850871858     
m=63: N = 120438826089053357522       
m=64: N =  54118206965024458802       


  


4.Reference

[1]: Duncan Moore, Taxicab Numbers and Cabtaxi Numbers