[ October 23, 2015 ]
Expressing the natural numbers as an absolute difference
between two powers

Here I am trying to collect the first natural numbers
expressed as differences between two powers
with base and exponent greater than or equal to 2.
I have no clue as to 'in how many different ways' each
natural number N can be expressed in that manner.
Nor have I knowledge if all natural numbers N have solutions.

B.S. Rangaswamy soon communicated me the following: All odd numbers, happen to be differences of adjacent squares as in: 29 = 152 - 142 43 = 222 - 212 59 = 302 - 292 67 = 342 - 332 69 = 352 - 342 All even numbers evenly divisible by 4, happen to be differences of alternate squares as in: 68 = 182 - 162 72 = 192 - 172

Note for instance that 2⁴ = 4² or that 2⁶ = 4³ = 8²
In the table I use the following notation and it means
choose any one from the powers between brackets.
Twofold cases
16 = [2⁴][4²], 81 = [3⁴][9²], 512 = [2⁹][8³], 625 = [5⁴][25²],
1296 = [6⁴][36²], 2401 = [7⁴][49²], 19683 = [3⁹][27³],
50625 = [15⁴][225²], 279841 = [23⁴][529²],
1953125 = [5⁹][125³], 10077696 = [6⁹][216³], etc.
Threefold cases
64 = [2⁶][4³][8²], 256 = [2⁸][4⁴][16²], 729 = [3⁶][9³][27²],
1024 = [2¹⁰][4⁵][32²], 6561 = [3⁸][9⁴][81²], 15625 = [5⁶][25³][125²],
16384 = [2¹⁴][4⁷][128²], 32768 = [2¹⁵][8⁵][32³],
390625 = [5⁸][25⁴][625²], etc.
Fourfold cases
65536 = [2¹⁶][4⁸][16⁴][256²], etc.
Fivefold cases
4096 = [2¹²][4⁶][8⁴][16³][64²], 262144 = [2¹⁸][4⁹][8⁶][64³][512²], etc.

Let me see how far I can go with numbers N up to 100.

Here is the sequence with 12 leftovers
6, 14, 34, 42, 50, 58, 62, 66, 70, 78, 82, 86, 90
Help me narrow it down or prove it is unsolvable.

Number N	(b1)(p1) - (b2)(p2)	expansions
1	32 - 23	9 - 8
2	33 - 52	27 - 25
3	27 - 53	128 - 125
4	23 - 22 62 - 25 53 - 112	8 - 4  36 - 25  125 - 121
5	32 - 22 25 - 33	9 - 4  32 - 27
6	?
7	[24][42] - 32 25 - 52 27 - 112 [215][85][323] - 1812	16 - 9  32 - 25  128 - 121  32768 - 32761
8	[24][42] - 23 3122 - 463	16 - 8  97344 - 97336
9	52 - [24][42] 62 - 33 152 - 63 2532 - 403	25 - 16  36 - 27  225 - 216  64009 - 64000
10	133 - 37	2197 - 2187
11	33 - [24][42] 62 - 52 562 - 55 153 - 582	27 - 16  36 - 25  3136 - 3125  3375 - 3364
12	[24][42] - 22 472 - 133	16 - 4  2209 - 2197
13	72 - 62 [28][44][162] - 35 173 - 702	49 - 36  256 - 243  4913 - 4900
14	?
15	[26][43][82] - 72 11382 - 1093	64 - 49  1295044 - 1295029
16	25 - [24][42] 52 - 32 122 - 27	32 - 16  25 - 9  144 - 128
17	52 - 23 72 - 25 [34][92] - [26][43][82] 232 - [29][83] 2822 - 433 3752 - 523 3786612 - 52343	25 - 8  49 - 32  81 - 64  529 - 512  79524 - 79507  140625 - 140608  143384152921 - 143384152904
18	33 - 32 35 - 152 192 - 73	27 - 9  243 - 225  361 - 343
19	33 - 23 102 - [34][92] 122 - 53 73 - 182 555 - 224342	27 - 8  100 - 81  144 - 125  343 - 324  503284375 - 503284356
20	62 - [24][42] 63 - 142	36 - 16  216 - 196
21	52 - 22 112 - 102	25 - 4  121 - 100
22	72 - 33 472 - 37	49 - 27  2209 - 2187
23	33 - 22 25 - 32 122 - 112 211 - 452	27 - 4  32 - 9  144 - 121  2048 - 2025
24	25 - 23 72 - 52 [210][45][322] - 103 7368442 - 81583	32 - 8  49 - 25  1024 - 1000  542939080336 - 542939080312
25	53 - 102 132 - 122	125 - 100  169 - 144
26	353 - 2072 25372 - 235	42875 - 42849  6436369 - 6436343
27	62 - 32 142 - 132 35 - 63	36 - 9  196 - 169  243 - 216
28	25 - 22 62 - 23 [26][43][82] - 62 27 - 102 [29][83] - 222 373 - [154][2252] 217 - 3622	32 - 4  36 - 8  64 - 36  128 - 100  512 - 484  50653 - 50625  131072 - 131044
29	152 - 142	225 - 196
30	832 - 193	6889 - 6859
31	[28][44][162] - 152	256 - 225
32	62 - 22 [26][43][82] - 25 [34][92] - 72 65 - 882	36 - 4  64 - 32  81 - 49  7776 - 7744
33	72 - [24][42] 172 - [28][44][162]	49 - 16  289 - 256
34	?
35	182 - 172 113 - [64][362]	324 - 289  1331 - 1296
36	102 - [26][43][82] 422 - 123	100 - 64  1764 - 1728
37	[26][43][82] - 33 192 - 182 37882 - [315][275][2433]	64 - 27  361 - 324  14348944 - 14348907
38	372 - 113	1369 - 1331
39	[26][43][82] - 52 202 - 192 103 - 312 223 - 1032	64 - 25  400 - 361  1000 - 961  10648 - 10609
40	72 - 32 112 - [34][92] [28][44][162] - 63 143 - 522	49 - 9  121 - 81  256 - 216  2744 - 2704
41	72 - 23 132 - 27 212 - 202	49 - 8  169 - 128  441 - 400
42	?
43	222 - 212	484 - 441
44	53 - [34][92] 122 - 102 132 - 53	125 - 81  144 - 100  169 - 125
45	72 - 22 [34][92] - 62 232 - 222 213 - 962	49 - 4  81 - 36  529 - 484  9261 - 9216
46	172 - 35	289 - 243
47	27 - [34][92] 63 - 132 35 - 142 242 - 232 123 - 412 633 - 5002	128 - 81  216 - 169  243 - 196  576 - 529  1728 - 1681  250047 - 250000
48	[26][43][82] - [24][42] 132 - 112 283 - 1482	64 - 16  169 - 121  21952 - 21904
49	[34][92] - 25 [54][252] - 242 653 - 5242	81 - 32  625 - 576  274625 - 274576
50	?
51	102 - 72 262 - [54][252]	100 - 49  676 - 625
52	142 - 122	196 - 144
53	[36][93][272] - 262 293 - 1562	729 - 676  24389 - 24336
54	[34][92] - 33 73 - 172	81 - 27  343 - 289
55	[26][43][82] - 32 282 - [36][93][272] 563 - 4192	64 - 9  784 - 729  175616 - 175561
56	[26][43][82] - 23 [34][92] - 52 152 - 132 183 - 762	64 - 8  81 - 25  225 - 169  5832 - 5776
57	112 - [26][43][82] 202 - 73 292 - 282	121 - 64  400 - 343  841 - 784
58	?
59	302 - 292	900 - 841
60	[26][43][82] - 22 [28][44][162] - 142 1363 - 15862 765 - 503542	64 - 4  256 - 196  2515456 - 2515396  2535525376 - 2535525316
61	53 - [26][43][82] 312 - 302	125 - 64  961 - 900
62	?
63	122 - [34][92] [210][45][322] - 312 5683 - 135372	144 - 81  1024 - 961  183250432 - 183250369
64	102 - 62 27 - [26][43][82] 172 - 152 242 - [29][83]	100 - 36  128 - 64  289 - 225  576 - 512
65	[34][92] - [24][42] 332 - [210][45][322] 532 - 143 141132 - 5843	81 - 16  1089 - 1024  2809 - 2744  199176769 - 199176704
66	?
67	342 - 332 233 - 1102	1156 - 1089  12167 - 12100
68	102 - 25 142 - 27 182 - [28][44][162] 462 - 211 18742 - 1523	100 - 32  196 - 128  324 - 256  2116 - 2048  3511876 - 3511808
69	132 - 102 352 - 342	169 - 100  1225 - 1156
70	?
71	142 - 53 [29][83] - 212 [64][362] - 352 37 - 462	196 - 125  512 - 441  1296 - 1225  2187 - 2116
72	[34][92] - 32 112 - 72 63 - 122 192 - 172	81 - 9  121 - 49  216 - 144  361 - 289
73	[34][92] - 23 102 - 33 172 - 63 372 - [64][362] 6112 - 723 67172 - 3563	81 - 8  100 - 27  289 - 216  1369 - 1296  373321 - 373248  45118089 - 45118016
74	35 - 132 993 - 9852	243 - 169  970299 - 970225
75	102 - 52 142 - 112 382 - 372	100 - 25  196 - 121  1444 - 1369
76	53 - 72 202 - 182 1013 - 10152	125 - 49  400 - 324  1030301 - 1030225
77	[34][92] - 22 392 - 382	81 - 4  1521 - 1444
78	?
79	27 - 72 402 - 392 203 - 892 3022 - 453	128 - 49  1600 - 1521  8000 - 7921  91204 - 91125
80	122 - [26][43][82] 212 - 192 2922 - 443	144 - 64  441 - 361  85264 - 85184
81	152 - 122 182 - 35 412 - 402 133 - 462	225 - 144  324 - 243  1681 - 1600  2197 - 2116
82	?
83	422 - 412 [39][273] - 1402	1764 - 1681  19683 - 19600
84	102 - [24][42] 222 - 202	100 - 16  484 - 400
85	112 - 62 432 - 422	121 - 36  1849 - 1764
86	?
87	[28][44][162] - 132 73 - [28][44][162] 442 - 432	256 - 169  343 - 256  1936 - 1849
88	132 - [34][92] 63 - 27 232 - 212	169 - 81  216 - 128  529 - 441
89	112 - 25 53 - 62 332 - 103 452 - 442 912 - 213 4082 - 553	121 - 32  125 - 36  1089 - 1000  2025 - 1936  8281 - 8192  166464 - 166375
90	?
91	102 - 32 63 - 53 462 - 452	100 - 9  216 - 125  2116 - 2025
92	102 - 23 27 - 62 242 - 222 213 - 902	100 - 8  128 - 36  576 - 484  8192 - 8100
93	53 - 25 172 - 142 472 - 462 1302 - 75	125 - 32  289 - 196  2209 - 2116  16900 - 16807
94	112 - 33 4212 - 311	121 - 27  177241 - 177147
95	63 - 112 122 - 72 482 - 472 67 - [234][5292]	216 - 121  144 - 49  2304 - 2209  279936 - 279841
96	102 - 22 112 - 52 27 - 25 142 - 102 [54][252] - 232	100 - 4  121 - 25  128 - 32  196 - 100  625 - 529
97	152 - 27 [74][492] - 482 772 - 183	225 - 128  2401 - 2304  5929 - 5832
98	53 - 33 212 - 73	125 - 27  441 - 343
99	35 - 122 182 - 152 502 - [74][492]	243 - 144  324 - 225  2500 - 2401
100	53 - 52 152 - 53 73 - 35 262 - 242 103 - 302 55 - 552 902 - 203 1182 - 243 343 - 1982 1371902 - 26603	125 - 25  225 - 125  343 - 243  676 - 576  1000 - 900  3125 - 3025  8100 - 8000  13924 - 13824  39304 - 39204  18821096100 - 18821096000

An analogue problem with the sum of three cubes
The Uncracked Problem with 33 - Numberphile

More info (e.g. state of affairs) from the OEIS database
A066510 and A074981