The nth Taxicab number is the smallest integer that can be written as the sum of two positive cubes in n different ways. It is known that there are numbers expressible in as the sum of two cubes in any number of ways.

Can we extend the concept of Taxicab numbers to higher powers? Jaroslaw Wroblewski and Stuart Gascoigne proved that for all powers, there are numbers expressible as the sum of (n+1) nth powers in m ways, for any value of m. (see proof).Can we use less than n+1 powers? Can we find 3-way or 4-way sums even if m-way sums do not exist? I will use the notation Taxicab(n,p,m) to denote the smallest number expressible in m ways as the sum of p nth powers.

The obvious first step is to try and find numbers expressible as the sum of two fourth powers in several ways. The smallest number expressible as the sum of two fourth powers in two ways is

635318657 = 133⁴ + 134⁴ = 158⁴ + 59⁴

found by Euler.

If there is a number expressible as the sum of two fourth powers in three ways it is Taxicab(4,2,3) > 2.6x10²⁶ (Stuart Gascoigne).

Can we use sums of three fourth powers? The answer is yes. Hardy and Wright prove that there are numbers expressible as the sums of three fourth powers in any number of ways.

Below are listed the first 18 Taxicab(4,3,m) numbers. Interestingly, we can note that Taxicab(4,3,5) includes a zero term. Also note that the series goes from 10 to 12 to 16 to 18, missing out the odd terms.

Taxicab(4,3,1) = 1
= 1⁴ + 0⁴ + 0⁴

Taxicab(4,3,2) = 2673
= 7⁴ + 4⁴ + 2⁴
= 6⁴ + 6⁴ + 3⁴

Taxicab(4,3,3) = 811538
= 29⁴ + 17⁴ + 12⁴
= 28⁴ + 21⁴ + 7⁴
= 27⁴ + 23⁴ + 4⁴

Taxicab(4,3,4) = 5978882
= 48⁴ + 25⁴ + 23⁴
= 47⁴ + 32⁴ + 15⁴
= 45⁴ + 37⁴ + 8⁴
= 43⁴ + 40⁴ + 3⁴

Taxicab(4,3,5) = 137149922
= 105⁴ + 56⁴ + 49⁴
= 104⁴ + 65⁴ + 39⁴
= 99⁴ + 80⁴ + 19⁴
= 96⁴ + 85⁴ + 11⁴
= 91⁴ + 91⁴ + 0⁴

Taxicab(4,3,6) = 292965218
= 127⁴ + 66⁴ + 61⁴
= 126⁴ + 77⁴ + 49⁴
= 123⁴ + 89⁴ + 34⁴
= 121⁴ + 94⁴ + 27⁴
= 119⁴ + 98⁴ + 21⁴
= 111⁴ + 109⁴ + 2⁴

Taxicab(4,3,7) = 779888018
= 167⁴ + 38⁴ + 9⁴
= 162⁴ + 89⁴ + 73⁴
= 161⁴ + 98⁴ + 63⁴
= 158⁴ + 111⁴ + 47⁴
= 157⁴ + 114⁴ + 43⁴
= 147⁴ + 133⁴ + 14⁴
= 142⁴ + 139⁴ + 3⁴

Taxicab(4,3,8) = 5745705602
= 267⁴ + 145⁴ + 122⁴
= 265⁴ + 163⁴ + 102⁴
= 263⁴ + 173⁴ + 90⁴
= 262⁴ + 177⁴ + 85⁴
= 255⁴ + 197⁴ + 58⁴
= 250⁴ + 207⁴ + 43⁴
= 243⁴ + 218⁴ + 25⁴
= 233⁴ + 230⁴ + 2⁴

Taxicab(4,3,10) = 49511121842
= 458⁴ + 233⁴ + 225⁴
= 457⁴ + 255⁴ + 202⁴
= 455⁴ + 273⁴ + 182⁴
= 447⁴ + 310⁴ + 137⁴
= 442⁴ + 325⁴ + 117⁴
= 441⁴ + 322⁴ + 175⁴
= 438⁴ + 335⁴ + 103⁴
= 427⁴ + 357⁴ + 70⁴
= 413⁴ + 378⁴ + 35⁴
= 403⁴ + 390⁴ + 13⁴

Taxicab(4,3,12) = 281539574498
= 707⁴ + 371⁴ + 336⁴
= 704⁴ + 411⁴ + 293⁴
= 701⁴ + 432⁴ + 269⁴
= 699⁴ + 443⁴ + 256⁴
= 696⁴ + 457⁴ + 239⁴
= 693⁴ + 469⁴ + 224⁴
= 679⁴ + 511⁴ + 168⁴
= 664⁴ + 543⁴ + 121⁴
= 656⁴ + 557⁴ + 99⁴
= 649⁴ + 568⁴ + 81⁴
= 632⁴ + 591⁴ + 41⁴
= 616⁴ + 609⁴ + 7⁴

Taxicab(4,3,16) = 7865870969138
= 1626⁴ + 829⁴ + 797⁴
= 1622⁴ + 911⁴ + 711⁴
= 1621⁴ + 922⁴ + 699⁴
= 1619⁴ + 941⁴ + 678⁴
= 1611⁴ + 997⁴ + 614⁴
= 1601⁴ + 1047⁴ + 554⁴
= 1579⁴ + 1126⁴ + 453⁴
= 1578⁴ + 1129⁴ + 449⁴
= 1571⁴ + 1149⁴ + 422⁴
= 1553⁴ + 1194⁴ + 359⁴
= 1549⁴ + 1203⁴ + 346⁴
= 1503⁴ + 1289⁴ + 214⁴
= 1494⁴ + 1303⁴ + 191⁴
= 1481⁴ + 1322⁴ + 159⁴
= 1447⁴ + 1366⁴ + 81⁴
= 1434⁴ + 1381⁴ + 53⁴

Taxicab(4,3,18) = 47580188090162
= 2548⁴ + 1365⁴ + 1183⁴
= 2545⁴ + 1413⁴ + 1132⁴
= 2540⁴ + 1467⁴ + 1073⁴
= 2535⁴ + 1508⁴ + 1027⁴
= 2525⁴ + 1572⁴ + 953⁴
= 2523⁴ + 1583⁴ + 940⁴
= 2515⁴ + 1623⁴ + 892⁴
= 2492⁴ + 1715⁴ + 777⁴
= 2487⁴ + 1732⁴ + 755⁴
= 2457⁴ + 1820⁴ + 637⁴
= 2443⁴ + 1855⁴ + 588⁴
= 2405⁴ + 1937⁴ + 468⁴
= 2372⁴ + 1997⁴ + 375⁴
= 2353⁴ + 2028⁴ + 325⁴
= 2327⁴ + 2067⁴ + 260⁴
= 2303⁴ + 2100⁴ + 203⁴
= 2293⁴ + 2113⁴ + 180⁴
= 2212⁴ + 2205⁴ + 7⁴

Taxicab(4,3,19) = 101635786393778
= 3173⁴ + 722⁴ + 171⁴
= 3083⁴ + 1542⁴ + 1541⁴
= 3082⁴ + 1609⁴ + 1473⁴
= 3078⁴ + 1691⁴ + 1387⁴
= 3066⁴ + 1813⁴ + 1253⁴
= 3059⁴ + 1862⁴ + 1197⁴
= 3037⁴ + 1978⁴ + 1059⁴
= 3017⁴ + 2058⁴ + 959⁴
= 3002⁴ + 2109⁴ + 893⁴
= 2983⁴ + 2166⁴ + 817⁴
= 2967⁴ + 2209⁴ + 758⁴
= 2957⁴ + 2234⁴ + 723⁴
= 2934⁴ + 2287⁴ + 647⁴
= 2891⁴ + 2373⁴ + 518⁴
= 2863⁴ + 2422⁴ + 441⁴
= 2793⁴ + 2527⁴ + 266⁴
= 2763⁴ + 2566⁴ + 197⁴
= 2727⁴ + 2609⁴ + 118⁴
= 2698⁴ + 2641⁴ + 57⁴

found by Duncan Moore

Taxicab(4,3,24) = 385427677487762
= 4301⁴ + 2241⁴ + 2060⁴
= 4300⁴ + 2271⁴ + 2029⁴
= 4295⁴ + 2364⁴ + 1931⁴
= 4291⁴ + 2415⁴ + 1876⁴
= 4285⁴ + 2476⁴ + 1809⁴
= 4284⁴ + 2485⁴ + 1799⁴
= 4260⁴ + 2651⁴ + 1609⁴
= 4249⁴ + 2709⁴ + 1540⁴
= 4219⁴ + 2839⁴ + 1380⁴
= 4196⁴ + 2921⁴ + 1275⁴
= 4179⁴ + 2975⁴ + 1204⁴
= 4164⁴ + 3019⁴ + 1145⁴
= 4155⁴ + 3044⁴ + 1111⁴
= 4125⁴ + 3121⁴ + 1004⁴
= 4100⁴ + 3179⁴ + 921⁴
= 4081⁴ + 3220⁴ + 861⁴
= 4049⁴ + 3284⁴ + 765⁴
= 3955⁴ + 3444⁴ + 511⁴
= 3941⁴ + 3465⁴ + 476⁴
= 3916⁴ + 3501⁴ + 415⁴
= 3869⁴ + 3564⁴ + 305⁴
= 3836⁴ + 3605⁴ + 231⁴
= 3831⁴ + 3611⁴ + 220⁴
= 3756⁴ + 3695⁴ + 61⁴

found by Duncan Moore

(This is sequence A085559 from the Online Encyclopedia of Integer Sequences)

References

G.H. Hardy & E.M. Wright: An introduction to the theory of numbers 5th edition (1979) p330