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The Nine Digits Page 9
with some Ten Digits (pandigital) exceptions
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When I use the term ninedigital in these articles I always refer to a strictly zeroless pandigital (digits from 1 to 9 each appearing just once).

Ninth Page

  Topic 9.1   [ December 22, 2025 ]
From Henry E. Dudeney's book “Amusements In Mathematics” (puzzle 92)
Arranging the \(9\) and \(10\) digits of the nine- and pandigital numbers resulting in concatenations of 1, 2, 3, 4, ... squares.

Dudeney just gave one example namely \(981324576\) where the nine digits are so arranged that they form "four"
square numbers: \([9][81][324][576]\) or \([3^2][9^2][18^2][24^2]\).
Evidently the permutations of these four squares also result in ninedigitals.
Can you create the full list of all solutions ?

What about other than four squares? How many arrangements can you find that gives "five', "three" and "two" squares?
It is a similar challenge as with the four squares. It shouldn't be so difficult to squeak your program code in order
to find these e.g. "three" or "two" solutions as well.

"Now, can you put the nine digits all together so they form a "single" square number?" Dudeney asked.
The smallest possible and largest possible solutions are already studied and known. Follow this link to Topic 2.2
\([139854276]\) is \([11826^2]~~\) and \(~~[923187456]\) is \([30384^2]\)

And now it is time to delve into the many possible variations.
Change the 'squares' into 'primes', 'cubes' or 'higher powers'. Oh yes, what would be the largest power possible.
What about a mix of powers. What about doubling or tripling the digits of the ninedigitals and pandigitals...
Be as creative as possible to the best of your ability. Success!

 

The analogue challenge now with pandigitals. Provide the full listings.
Here are some examples to get you going.

Four squares→ \([4][25][81][30976]\) is \([2^2][5^2][9^2][176^2]\)

Three squares→ \([9][81][4730625]\) is \([3^2][9^2][2175^2]\)

Two squares→ \([7056][321489]\) is \([84^2][567^2]\)

One square → \([1026753849]\) is \([32043^2]~~\) and \(~~[9814072356]\) is \([99066^2]\)

[ January 10-18, 2026 ]
Alexandru Petrescu came to the rescue. He provided me a list of all the solutions with two, three, four and five factors.
And even one mixed powers solution with six factors!
Note : the factor \(1\) will only be considered with exponent \(2\).

NINEDIGITAL SOLUTIONS
TWO FACTORS
EQUAL POWERS [n^2]
[324][751689] = [182][8672]
[3249][15876] = [572][1262]
[36][5184729] = [62][22772]
[36][5948721] = [62][24392]
[4356][71289] = [662][2672]
[576][321489] = [242][5672]
[576][349281] = [242][5912]
[576][381924] = [242][6182]
[729][385641] = [272][6212]
[81][2537649] = [92][15932]
[81][5673924] = [92][23822]
[81][7436529] = [92][27272]
[81][9253764] = [92][30422]
[8649][35721] = [932][1892]
[9][13527684] = [32][36782]
[9][34857216] = [32][59042]
[9][65318724] = [32][80822]
[9][73256481] = [32][85592]
[9][81432576] = [32][90242]


TWO FACTORS
EQUAL POWERS [n^3]
[125][438976] = [53][763]
[512][438976] = [83][763]
[8][24137569] = [23][2893]
[8][32461759] = [23][3193]
  TWO FACTORS
MIXED POWERS
[128][379456] = [37][6162]
[1369][42875] = [372][353]
[1369][54872] = [372][383]
[169][537824] = [132][145]
[1728][34596] = [123][1862]
[1728][45369] = [123][2132]
[1936][42875] = [442][353]
[1936][54872] = [442][383]
[196][537824] = [142][145]
[2187][34596] = [37][1862]
[2187][45369] = [37][2132]
[243][751689] = [35][8672]
[27][1354896] = [33][11642]
[27][3481956] = [33][18662]
[27][3594816] = [33][18962]
[27][5391684] = [33][23222]
[27][6395841] = [33][25292]
[32][4519876] = [25][21262]
[4][79235168] = [22][385]
[512][438976] = [29][763]
[5832][47961] = [183][2192]
[729][385641] = [36][6212]
[729][385641] = [93][6212]
[7569][13824] = [872][243]
[81][2537649] = [34][15932]
[8][13942756] = [23][37342]
[81][5673924] = [34][23822]
[81][7436529] = [34][27272]
[81][9253764] = [34][30422]
[8][21473956] = [23][46342]
[8][21743569] = [23][46632]
[8][23174596] = [23][48142]
[8][24137569] = [23][176]
[8][24137569] = [23][49132]
[8][52374169] = [23][72372]
[8][54213769] = [23][73632]
[8][74356129] = [23][86232]
[961][537824] = [312][145]
  THREE FACTORS
EQUAL POWERS
[1][25][438976] = [12][52][7632]
[1][256][73984] = [12][162][2722]
[1][4][3297856] = [12][22][18162]
[1][4][3857296] = [12][22][19642]
[1][4][5827396] = [12][22][24142]
[1][4][6385729] = [12][22][25272]
[1][4][8567329] = [12][22][29272]
[1][4][9572836] = [12][22][30942]
[1][49][872356] = [12][72][9342]
[1][576][24389] = [12][242][2932]
[16][25][73984] = [42][52][2722]
[16][784][5329] = [42][282][732]
[1][64][537289] = [12][82][7332]
[1][625][73984] = [12][252][2722]
[25][784][1369] = [52][282][372]
[25][784][1936] = [52][282][442]
[25][841][7396] = [52][292][862]
[361][529][784] = [192][232][282]
[36][729][5184] = [62][272][722]
[36][81][74529] = [62][92][2732]
[36][81][79524] = [62][92][2822]
[4][16][537289] = [22][42][7332]
[4][25][139876] = [22][52][3742]
[4][25][391876] = [22][52][6262]
[4][289][15376] = [22][172][1242]
[81][324][7569] = [92][182][872]
[81][576][3249] = [92][242][572]
[81][729][4356] = [92][272][662]
[9][324][15876] = [32][182][1262]
  THREE FACTORS
MIXED POWERS
[1][625][73984] = [12][54][2722]
[16][25][73984] = [24][52][2722]
[1][64][537289] = [12][26][7332]
[1][64][537289] = [12][43][7332]
[16][784][5329] = [24][282][732]
[1][8][2537649] = [12][23][15932]
[1][8][5673924] = [12][23][23822]
[1][8][7436529] = [12][23][27272]
[1][8][9253764] = [12][23][30422]
[1][9][8365427] = [12][32][2033]
[27][36][51984] = [33][62][2282]
[27][36][95481] = [33][62][3092]
[27][81][34596] = [33][34][1862]
[27][81][45369] = [33][34][2132]
[32][49][15876] = [25][72][1262]
[32][576][1849] = [25][242][432]
[32][81][45796] = [25][34][2142]
[32][841][7569] = [25][292][872]
[36][49][78125] = [62][72][57]
[36][729][5184] = [62][36][722]
[36][729][5184] = [62][93][722]
[36][81][74529] = [62][34][2732]
[36][81][79524] = [62][34][2822]
[4][16][537289] = [22][24][7332]
[4][32][751689] = [22][25][8672]
[4][3721][6859] = [22][612][193]
[81][243][7569] = [34][35][872]
[81][243][7569] = [92][35][872]
[81][324][7569] = [34][182][872]
[81][576][3249] = [34][242][572]
[81][729][4356] = [34][36][662]
[81][729][4356] = [92][36][662]
[81][729][4356] = [34][93][662]
[81][729][4356] = [92][93][662]
[81][729][4356] = [34][272][662]
[8][1764][5329] = [23][422][732]
[8][2197][4356] = [23][133][662]
[8][243][17956] = [23][35][1342]
[8][27][195364] = [23][33][4422]
[8][27][351649] = [23][33][5932]
[8][27][395641] = [23][33][6292]
[8][27][543169] = [23][33][7372]
[8][324][17956] = [23][182][1342]
[8][361][74529] = [23][192][2732]
[8][361][79524] = [23][192][2822]
[8][4356][7921] = [23][662][892]
[8][4761][5329] = [23][692][732]
[8][576][13924] = [23][242][1182]
[8][729][13456] = [23][36][1162]
[8][729][13456] = [23][93][1162]
[8][729][13456] = [23][272][1162]
[9][16][537824] = [32][24][145]
[9][16][537824] = [32][42][145]
[9][1728][4356] = [32][123][662]
[9][1764][5832] = [32][422][183]
[9][2187][4356] = [32][37][662]
[9][243][15876] = [32][35][1262]
[9][27][385641] = [32][33][6212]
[9][32][758641] = [32][25][8712]
  FOUR FACTORS
EQUAL POWERS
[1][36][529][784] = [12][62][232][282]
[1][4][9][872356] = [12][22][32][9342]
[4][25][81][7396] = [22][52][92][862]
[9][25][361][784] = [32][52][192][282]
[9][81][324][576] = [32][3^4][182][242]


FOUR FACTORS
MIXED POWERS
[1][8][243][7569] = [12][23][35][872]
[1][8][27][34596] = [12][23][33][1862]
[1][8][27][45369] = [12][23][33][2132]
[1][8][32][45796] = [12][23][25][2142]
[1][8][324][7569] = [12][23][182][872]
[1][8][36][74529] = [12][23][62][2732]
[1][8][36][79524] = [12][23][62][2822]
[1][8][576][3249] = [12][23][242][572]
[1][8][729][4356] = [12][23][36][662]
[1][8][729][4356] = [12][23][93][662]
[1][8][729][4356] = [12][23][272][662]
[1][9][36][42875] = [12][32][62][353]
[1][9][36][54872] = [12][32][62][383]
[32][49][81][576] = [25][72][34][242]
[32][49][81][576] = [25][72][92][242]
[4][25][81][7396] = [22][52][34][862]
[4][32][81][7569] = [22][25][34][872]
[4][32][81][7569] = [22][25][92][872]
[8][9][27][13456] = [23][32][33][1162]
[4][8][125][7396] = [22][23][53][862]
[4][8][32][17956] = [22][23][25][1342]
[4][8][512][7396] = [22][23][29][862]
[4][8][512][7396] = [22][23][83][862]
[4][9][32][15876] = [22][32][25][1262]
[4][9][36][78125] = [22][32][62][57]
[8][9][64][35721] = [23][32][26][1892]
[8][9][64][35721] = [23][32][43][1892]
[8][9][64][35721] = [23][32][82][1892]
[9][27][36][5184] = [32][33][62][722]
[9][27][81][4356] = [32][33][34][662]
[9][27][81][4356] = [32][33][92][662]
[9][32][576][841] = [32][25][242][292]
[9][32][81][5476] = [32][25][34][742]
[9][32][81][5476] = [32][25][92][742]
[9][36][125][784] = [32][62][53][282]
[9][36][512][784] = [32][62][29][282]
[9][81][243][576] = [32][34][35][242]
[9][81][243][576] = [32][92][35][242]
[9][81][324][576] = [32][34][182][242]
  FIVE FACTORS
EQUAL POWERS
[1][9][25][36][784] = [12][32][52][62][282]


FIVE FACTORS
MIXED POWERS
[1][4][8][25][7396] = [12][22][23][52][862]
[1][4][8][32][7569] = [12][22][23][25][872]
[1][8][32][49][576] = [12][23][25][72][242]
[1][8][9][243][576] = [12][23][32][35][242]
[1][8][9][27][4356] = [12][23][32][33][662]
[1][8][9][32][5476] = [12][23][32][25][742]
[1][8][9][324][576] = [12][23][32][182][242]
[4][9][32][81][576] = [22][32][25][34][242]
[4][9][32][81][576] = [22][32][25][92][242]


SIX FACTORS
MIXED POWERS
[1][4][8][9][32][576] = [12][22][23][32][25][242]


PANDIGITAL SOLUTIONS
TWO FACTORS
EQUAL POWERS [n^2]
[15876][23409] = [1262][1532]
[15876][39204] = [1262][1982]
[1764][893025] = [422][9452]
[20736][51984] = [1442][2282]
[20736][95481] = [1442][3092]
[2304][751689] = [482][8672]
[30276][51984] = [1742][2282]
[30276][95481] = [1742][3092]
[324][6517809] = [182][25532]
[324][8590761] = [182][29312]
[3249][576081] = [572][7592]
[36][19847025] = [62][44552]
[36][57108249] = [62][75572]
[36][71520849] = [62][84572]
[36][80514729] = [62][89732]
[36][94187025] = [62][97052]
[38025][47961] = [1952][2192]
[4761][893025] = [692][9452]
[5184][207936] = [722][4562]
[5184][603729] = [722][7772]
[576][1034289] = [242][10172]
[576][1432809] = [242][11972]
[576][2039184] = [242][14282]
[576][2893401] = [242][17012]
[576][3814209] = [242][19532]
[576][9108324] = [242][30182]
[63504][71289] = [2522][2672]
[7056][321489] = [842][5672]
[7056][349281] = [842][5912]
[7056][381924] = [842][6182]
[729][3048516] = [272][17462]
[729][4835601] = [272][21992]
[729][5308416] = [272][23042]
[7569][408321] = [872][6392]
[7569][423801] = [872][6512]
[81][32970564] = [92][57422]
[81][34257609] = [92][58532]
[81][45239076] = [92][67262]
[81][56972304] = [92][75482]
[81][70459236] = [92][83942]
[81][73925604] = [92][85982]
[81][74390625] = [92][86252]
[8649][731025] = [932][8552]
[9][102576384] = [32][101282]
[9][105637284] = [32][102782]
[9][158306724] = [32][125822]
[9][176305284] = [32][132782]
[9][180472356] = [32][134342]
[9][183467025] = [32][135452]
[9][187635204] = [32][136982]
[9][208571364] = [32][144422]
[9][218034756] = [32][147662]
[9][284057316] = [32][168542]
[9][307265841] = [32][175292]
[9][316057284] = [32][177782]
[9][430728516] = [32][207542]
[9][472801536] = [32][217442]
[9][475283601] = [32][218012]
[9][560837124] = [32][236822]
[9][570684321] = [32][238892]
[9][576432081] = [32][240092]
[9][734681025] = [32][271052]
[9][783104256] = [32][279842]
[9][825470361] = [32][287312]
[9][853107264] = [32][292082]


TWO FACTORS
EQUAL POWERS [n^3]
[9261][804357] = [213][933]
  TWO FACTORS
MIXED POWERS
[125][3748096] = [53][444]
[125][3748096] = [53][19362]
[125][6390784] = [53][25282]
[128][3904576] = [27][19762]
[1296][804357] = [362][933]
[1296][804357] = [64][933]
[1728][356409] = [123][5972]
[1764][328509] = [422][693]
[20736][51984] = [124][2282]
[20736][95481] = [124][3092]
[2187][356409] = [37][5972]
[243][6517809] = [35][25532]
[243][8590761] = [35][29312]
[24389][50176] = [293][2242]
[24389][51076] = [293][2262]
[27][30569841] = [33][552962]
[27][41563809] = [33][64472]
[27][63059481] = [33][79412]
[27][63154809] = [33][79472]
[2916][804357] = [542][933]
[3125][708964] = [55][8422]
[32][16785409] = [25][40972]
[32][56987401] = [25][75492]
[32][58079641] = [25][76212]
[32][58967041] = [25][76792]
[39601][42875] = [1992][353]
[39601][54872] = [1992][383]
[4][536870912] = [22][229]
[4761][328509] = [692][693]
[512][3748096] = [29][19362]
[512][3748096] = [29][444]
[512][6390784] = [29][25282]
[5832][104976] = [183][184]
[5832][104976] = [183][3242]
[5832][710649] = [183][8432]
[729][3048516] = [36][17462]
[729][3048516] = [93][17462]
[729][4835601] = [36][21992]
[729][4835601] = [93][21992]
[729][5308416] = [36][23042]
[729][5308416] = [93][23042]
[729][5308416] = [36][484]
[729][5308416] = [93][484]
[729][5308416] = [272][484]
[8][102495376] = [23][101242]
[8][104325796] = [23][102142]
[81][32970564] = [34][57422]
[81][34257609] = [34][58532]
[81][45239076] = [34][67262]
[81][56972304] = [34][75482]
[81][70459236] = [34][83942]
[81][73925604] = [34][85982]
[81][74390625] = [34][86252]
[8192][357604] = [213][5982]
[8][217356049] = [23][147432]
[8][235714609] = [23][153532]
[8][297631504] = [23][172522]
[8][403567921] = [23][200892]
[8][459630721] = [23][214392]
[8][491730625] = [23][221752]
[8][504271936] = [23][224562]
[8][534210769] = [23][231132]
[8][694375201] = [23][263512]
[8][793605241] = [23][281712]
[9216][804357] = [962][933]
  THREE FACTORS
EQUAL POWERS
[1][25][3748096] = [12][52][19362]
[1][25][6390784] = [12][52][25282]
[1][289][357604] = [12][172][5982]
[1][4][68973025] = [12][22][83052]
[1][4][85063729] = [12][22][92232]
[1][4][86397025] = [12][22][92952]
[1][49][7203856] = [12][72][26842]
[1][4][98327056] = [12][22][99162]
[1][49][8673025] = [12][72][29452]
[1][529][760384] = [12][232][8722]
[16][784][93025] = [42][282][3052]
[169][784][3025] = [132][282][552]
[1][784][390625] = [12][282][6252]
[1][784][635209] = [12][282][7972]
[196][784][3025] = [142][282][552]
[25][361][47089] = [52][192][2172]
[25][784][39601] = [52][282][1992]
[25][841][30976] = [52][292][1762]
[324][576][1089] = [182][242][332]
[324][576][9801] = [182][242][992]
[361][784][9025] = [192][282][952]
[36][81][257049] = [62][92][5072]
[36][81][497025] = [62][92][7052]
[36][81][725904] = [62][92][8522]
[4][1369][87025] = [22][372][2952]
[4][1936][87025] = [22][442][2952]
[4][25][3798601] = [22][52][19492]
[4][2809][15376] = [22][532][1242]
[4][289][173056] = [22][172][4162]
[4][3025][17689] = [22][552][1332]
[4][3025][18769] = [22][552][1372]
[4][3025][78961] = [22][552][2812]
[4][3025][81796] = [22][552][2862]
[4][5329][67081] = [22][732][2592]
[49][361][87025] = [72][192][2952]
[784][961][3025] = [282][312][552]
[81][2304][7569] = [92][482][872]
[81][3249][7056] = [92][572][842]
[81][576][23409] = [92][242][1532]
[81][576][39204] = [92][242][1982]
[81][729][63504] = [92][272][2522]
[9][1764][38025] = [32][422][1952]
[9][2304][15876] = [32][482][1262]
[9][324][576081] = [32][182][7592]
[9][36][7851204] = [32][62][28022]
[9][4761][38025] = [32][692][1952]
[9][5184][20736] = [32][722][1442]
[9][5184][30276] = [32][722][1742]
[9][576][408321] = [32][242][6392]
[9][576][423801] = [32][242][6512]
[9][6084][35721] = [32][782][1892]
[9][81][4730625] = [32][92][21752]
[9][81][6734025] = [32][92][25952]


  THREE FACTORS
MIXED POWERS
[1][25][3748096] = [12][52][444]
[16][784][93025] = [24][282][3052]
[1][7056][24389] = [12][842][293]
[1][784][390625] = [12][282][58]
[1][784][390625] = [12][282][254]
[1][8][32970564] = [12][23][57422]
[1][8][34257609] = [12][23][58532]
[1][8][45239076] = [12][23][67262]
[1][8][56972304] = [12][23][75482]
[1][8][70459236] = [12][23][83942]
[1][8][73925604] = [12][23][85982]
[1][8][74390625] = [12][23][86252]
[1][9][27543608] = [12][32][3023]
[243][576][1089] = [35][242][332]
[243][576][9801] = [35][242][992]
[25][64][389017] = [52][26][733]
[25][64][389017] = [52][43][733]
[25][64][389017] = [52][82][733]
[27][1089][4356] = [33][332][662]
[27][36][514089] = [33][62][7172]
[27][36][549081] = [33][62][7412]
[27][36][815409] = [33][62][9032]
[27][4356][9801] = [33][662][992]
[27][81][356409] = [33][34][5972]
[32][1089][5476] = [25][332][742]
[32][1849][7056] = [25][432][842]
[32][49][576081] = [25][72][7592]
[32][5476][9801] = [25][742][992]
[32][576][89401] = [25][242][2992]
[32][81][405769] = [25][34][6372]
[32][81][405769] = [25][92][6372]
[32][81][470596] = [25][34][6862]
[32][81][470596] = [25][92][6862]
[36][125][47089] = [62][53][2172]
[36][512][47089] = [62][29][2172]
[36][512][47089] = [62][83][2172]
[36][81][257049] = [62][34][5072]
[36][81][497025] = [62][34][7052]
[36][81][592704] = [62][34][843]
[36][81][592704] = [62][92][843]
[36][81][725904] = [62][34][8522]
[4][256][389017] = [22][28][733]
[4][256][389017] = [22][44][733]
[4][256][389017] = [22][162][733]
[4][32][6517809] = [22][25][25532]
[4][32][8590761] = [22][25][29312]
[4][5329][16807] = [22][732][75]
[4][625][389017] = [22][54][733]
[4][625][389017] = [22][252][733]
[64][81][205379] = [26][34][593]
[64][81][205379] = [26][92][593]
[64][81][205379] = [82][34][593]
[64][81][205379] = [82][92][593]
[81][2304][7569] = [34][482][872]
[8][1764][93025] = [23][422][3052]
[81][3249][7056] = [34][572][842]
[81][576][23409] = [34][242][1532]
[81][576][39204] = [34][242][1982]
[81][729][63504] = [34][36][2522]
[81][729][63504] = [34][93][2522]
[81][729][63504] = [34][272][2522]
[81][729][63504] = [92][36][2522]
[81][729][63504] = [92][93][2522]
[8][216][573049] = [23][63][7572]
[8][2197][63504] = [23][133][2522]
[8][2304][17956] = [23][482][1342]
[8][243][157609] = [23][35][3972]
[8][243][170569] = [23][35][4132]
[8][3025][47961] = [23][552][2192]
[8][324][157609] = [23][182][3972]
[8][324][170569] = [23][182][4132]
[8][3249][50176] = [23][572][2242]
[8][3249][51076] = [23][572][2262]
[8][36][1054729] = [23][62][10272]
[8][36][1974025] = [23][62][14052]
[8][361][257049] = [23][192][5072]
[8][36][2715904] = [23][62][16482]
[8][36][4507129] = [23][62][21232]
[8][361][497025] = [23][192][7052]
[8][36][5470921] = [23][62][23392]
[8][361][592704] = [23][192][843]
[8][361][725904] = [23][192][8522]
[8][36][7295401] = [23][62][27012]
[8][4096][35721] = [23][212][1892]
[8][4761][93025] = [23][692][3052]
[8][576][104329] = [23][242][3232]
[8][576][123904] = [23][242][3522]
[8][576][310249] = [23][242][5572]
[8][64][1750329] = [23][26][13232]
[8][64][1750329] = [23][82][13232]
[8][7056][13924] = [23][842][1182]
[8][729][341056] = [23][36][5842]
[8][7569][23104] = [23][872][1522]
[8][7569][32041] = [23][872][1792]
[8][7921][63504] = [23][892][2522]
[8][9][13675204] = [23][32][36982]
[8][9][17430625] = [23][32][41752]
[8][9][21307456] = [23][32][46162]
[8][9][35617024] = [23][32][59682]
[8][9][43507216] = [23][32][65962]
[8][9][56731024] = [23][32][75322]
[8][9604][35721] = [23][982][1892]
[9][125][760384] = [32][53][8722]
[9][128][357604] = [32][27][5982]
[9][1728][63504] = [32][243][2522]
[9][2187][63504] = [32][37][2522]
[9][2048][15376] = [32][211][1242]
[9][216][804357] = [32][63][933]
[9][243][576081] = [32][35][7592]
[9][27][3048516] = [32][33][17462]
[9][27][4835601] = [32][33][21992]
[9][27][5308416] = [32][33][23042]
[9][27][5308416] = [32][33][484]
[9][32][1048576] = [32][25][220]
[9][32][1056784] = [32][25][10282]
[9][32][1085764] = [32][25][10422]
[9][32][5740816] = [32][25][23962]
[9][32][5764801] = [32][25][78]
[9][32][6754801] = [32][25][25992]
[9][32][7845601] = [32][25][28012]
[9][5041][32768] = [32][712][215]
[9][512][760384] = [32][29][8722]
[9][5184][20736] = [32][722][124]
[9][7056][13824] = [32][842][243]
[9][81][2460375] = [32][34][1753]
[9][81][2460375] = [32][92][1753]
[9][81][4730625] = [32][34][21752]
[9][81][6734025] = [32][34][25952]
  FOUR FACTORS
EQUAL POWERS [n^2]
[1][25][36][47089] = [12][52][62][2172]
[1][36][49][87025] = [12][62][72][2952]
[1][36][784][9025] = [12][62][282][952]
[1][4][9][7203856] = [12][22][32][26842]
[1][4][9][8673025] = [12][22][32][29452]
[1][9][25][760384] = [12][32][52][8722]
[1][9][784][30625] = [12][32][282][1752]
[4][25][81][30976] = [22][52][92][1762]
[4][9][361][87025] = [22][32][192][2952]
[9][16][784][3025] = [32][42][282][552]
[9][81][324][7056] = [32][92][182][842]
[9][81][576][2304] = [32][92][242][482]


FOUR FACTORS
EQUAL POWERS [n^3]
[1][8][64][205379] = [13][23][43][593]


FOUR FACTORS
MIXED POWERS
[1][4][27][508369] = [12][22][33][7132]
[1][4][27][839056] = [12][22][33][9162]
[1][8][2304][7569] = [12][23][482][872]
[1][8][27][356409] = [12][23][33][5972]
[1][8][32][405769] = [12][23][25][6372]
[1][8][32][470596] = [12][23][25][6862]
[1][8][3249][7056] = [12][23][572][842]
[1][8][36][257049] = [12][23][62][5072]
[1][8][36][497025] = [12][23][62][7052]
[1][8][36][592704] = [12][23][62][843]
[1][8][36][725904] = [12][23][62][8522]
[1][8][576][23409] = [12][23][242][1532]
[1][8][576][39204] = [12][23][242][1982]
[1][8][64][205379] = [12][23][26][593]
[1][8][64][205379] = [12][23][82][593]
[1][8][729][63504] = [12][23][36][2522]
[1][8][729][63504] = [12][23][93][2522]
[1][8][729][63504] = [12][23][272][2522]
[1][8][9][2460375] = [12][23][32][1353]
[1][8][9][4730625] = [12][23][32][21752]
[1][8][9][6734025] = [12][23][32][25952]
[32][49][81][7056] = [25][72][34][842]
[32][49][81][7056] = [25][72][92][842]
[4][25][81][30976] = [22][52][34][1762]
[4][32][576][1089] = [22][25][242][332]
[4][32][576][9801] = [22][25][242][992]
[4][8][16][205379] = [22][23][24][593]
[4][8][16][205379] = [22][23][42][593]
[4][8][32][157609] = [22][23][25][3972]
[4][8][32][170569] = [22][23][25][4132]
[4][8][125][30976] = [22][23][53][1762]
[4][8][512][30976] = [22][23][29][1762]
[4][8][512][30976] = [22][23][83][1762]
[4][9][32][576081] = [22][32][25][7592]
[8][32][49][50176] = [23][25][72][2242]
[8][32][49][51076] = [23][25][72][2262]
[8][36][729][5041] = [23][62][36][712]
[8][36][729][5041] = [23][62][93][712]
[8][9][1764][3025] = [23][32][422][552]
[8][9][243][50176] = [23][32][35][2242]
[8][9][243][51076] = [23][32][35][2262]
[8][9][27][341056] = [23][32][33][5842]
[8][9][3025][4761] = [23][32][552][692]
[8][9][324][50176] = [23][32][182][2242]
[8][9][324][51076] = [23][32][182][2262]
[8][9][36][714025] = [23][32][62][8452]
[8][9][576][23104] = [23][32][242][1522]
[8][9][576][32041] = [23][32][242][1792]
[8][9][64][731025] = [23][32][26][8552]
[8][9][64][731025] = [23][32][43][8552]
[8][9][64][731025] = [23][32][82][8552]
[9][16][784][3025] = [32][24][282][552]
[9][27][81][63504] = [32][33][34][2522]
[9][27][81][63504] = [32][33][92][2522]
[9][32][841][7056] = [32][25][292][842]
[9][81][243][7056] = [32][34][35][842]
[9][81][324][7056] = [32][34][182][842]
[9][81][576][2304] = [32][34][242][482]
  FIVE FACTORS
EQUAL POWERS [n^2]
[1][4][9][36][87025] = [12][22][32][62][2952]


FIVE FACTORS
MIXED POWERS
[1][4][8][25][30976] = [12][22][23][52][1762]
[1][8][32][49][7056] = [12][23][25][72][842]
[1][8][9][243][7056] = [12][23][32][35][842]
[1][8][9][27][63504] = [12][23][32][33][2522]
[1][8][9][324][7056] = [12][23][32][182][842]
[1][8][9][576][2304] = [12][23][32][242][482]


SIX FACTORS
MIXED POWERS
[1][4][8][9][32][7056] = [12][22][23][32][25][842]


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E-mail address : pdg@worldofnumbers.com