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

Note with base and exponents greater than or equal to 2.

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.

It would be pretty cool to find the first number N that
withstands being written in that manner (for now this is N = 6).
Readers are invited to add to the table, to send reference
material, theorems or conjectures, links, and more ideas.

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
132 - 23 9 - 8 
233 - 52 27 - 25 
327 - 53 128 - 125 
423 - 22
62 - 25
53 - 112		 8 - 4 
36 - 25 
125 - 121 
532 - 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 
952 - [24][42]
62 - 33
152 - 63
2532 - 403		 25 - 16 
36 - 27 
225 - 216 
64009 - 64000 
10133 - 37 2197 - 2187 
1133 - [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 
1372 - 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 
1625 - [24][42]
52 - 32
122 - 27		 32 - 16 
25 - 9 
144 - 128 
1752 - 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 
1833 - 32
35 - 152
192 - 73		 27 - 9 
243 - 225 
361 - 343 
1933 - 23
102 - [34][92]
122 - 53
73 - 182
555 - 224342		 27 - 8 
100 - 81 
144 - 125 
343 - 324 
503284375 - 503284356 
2062 - [24][42]
63 - 142		 36 - 16 
216 - 196 
2152 - 22
112 - 102		 25 - 4 
121 - 100 
2272 - 33
472 - 37		 49 - 27 
2209 - 2187 
2333 - 22
25 - 32
122 - 112
211 - 452		 27 - 4 
32 - 9 
144 - 121 
2048 - 2025 
2425 - 23
72 - 52
[210][45][322] - 103
7368442 - 81583		 32 - 8 
49 - 25 
1024 - 1000 
542939080336 - 542939080312 
2553 - 102
132 - 122		 125 - 100 
169 - 144 
26353 - 2072
25372 - 235		 42875 - 42849 
6436369 - 6436343 
2762 - 32
142 - 132
35 - 63		 36 - 9 
196 - 169 
243 - 216 
2825 - 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 
29152 - 142 225 - 196 
30832 - 193 6889 - 6859 
31[28][44][162] - 152 256 - 225 
3262 - 22
[26][43][82] - 25
[34][92] - 72
65 - 882		 36 - 4 
64 - 32 
81 - 49 
7776 - 7744 
3372 - [24][42]
172 - [28][44][162]		 49 - 16 
289 - 256 
34?   
35182 - 172
113 - [64][362]		 324 - 289 
1331 - 1296 
36102 - [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 
38372 - 113 1369 - 1331 
39[26][43][82] - 52
202 - 192
103 - 312
223 - 1032		 64 - 25 
400 - 361 
1000 - 961 
10648 - 10609 
4072 - 32
112 - [34][92]
[28][44][162] - 63
143 - 522		 49 - 9 
121 - 81 
256 - 216 
2744 - 2704 
4172 - 23
132 - 27
212 - 202		 49 - 8 
169 - 128 
441 - 400 
42?   
43222 - 212 484 - 441 
4453 - [34][92]
122 - 102
132 - 53		 125 - 81 
144 - 100 
169 - 125 
4572 - 22
[34][92] - 62
232 - 222
213 - 962		 49 - 4 
81 - 36 
529 - 484 
9261 - 9216 
46172 - 35 289 - 243 
4727 - [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?   
51102 - 72
262 - [54][252]		 100 - 49 
676 - 625 
52142 - 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 
57112 - [26][43][82]
202 - 73
292 - 282		 121 - 64 
400 - 343 
841 - 784 
58?   
59302 - 292 900 - 841 
60[26][43][82] - 22
[28][44][162] - 142
1363 - 15862
765 - 503542		 64 - 4 
256 - 196 
2515456 - 2515396 
2535525376 - 2535525316 
6153 - [26][43][82]
312 - 302		 125 - 64 
961 - 900 
62?   
63122 - [34][92]
[210][45][322] - 312
5683 - 135372		 144 - 81 
1024 - 961 
183250432 - 183250369 
64102 - 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?   
67342 - 332
233 - 1102		 1156 - 1089 
12167 - 12100 
68102 - 25
142 - 27
182 - [28][44][162]
462 - 211
18742 - 1523		 100 - 32 
196 - 128 
324 - 256 
2116 - 2048 
3511876 - 3511808 
69132 - 102
352 - 342		 169 - 100 
1225 - 1156 
70?   
71142 - 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 
7435 - 132
993 - 9852		 243 - 169 
970299 - 970225 
75102 - 52
142 - 112
382 - 372		 100 - 25 
196 - 121 
1444 - 1369 
7653 - 72
202 - 182
1013 - 10152		 125 - 49 
400 - 324 
1030301 - 1030225 
77[34][92] - 22
392 - 382		 81 - 4 
1521 - 1444 
78?   
7927 - 72
402 - 392
203 - 892
3022 - 453		 128 - 49 
1600 - 1521 
8000 - 7921 
91204 - 91125 
80122 - [26][43][82]
212 - 192
2922 - 443		 144 - 64 
441 - 361 
85264 - 85184 
81152 - 122
182 - 35
412 - 402
133 - 462		 225 - 144 
324 - 243 
1681 - 1600 
2197 - 2116 
82?   
83422 - 412
[39][273] - 1402		 1764 - 1681 
19683 - 19600 
84102 - [24][42]
222 - 202		 100 - 16 
484 - 400 
85112 - 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 
88132 - [34][92]
63 - 27
232 - 212		 169 - 81 
216 - 128 
529 - 441 
89112 - 25
53 - 62
332 - 103
452 - 442
912 - 213
4082 - 553		 121 - 32 
125 - 36 
1089 - 1000 
2025 - 1936 
8281 - 8192 
166464 - 166375 
90?   
91102 - 32
63 - 53
462 - 452		 100 - 9 
216 - 125 
2116 - 2025 
92102 - 23
27 - 62
242 - 222
213 - 902		 100 - 8 
128 - 36 
576 - 484 
8192 - 8100 
9353 - 25
172 - 142
472 - 462
1302 - 75		 125 - 32 
289 - 196 
2209 - 2116 
16900 - 16807 
94112 - 33
4212 - 311		 121 - 27 
177241 - 177147 
9563 - 112
122 - 72
482 - 472
67 - [234][5292]		 216 - 121 
144 - 49 
2304 - 2209 
279936 - 279841 
96102 - 22
112 - 52
27 - 25
142 - 102
[54][252] - 232		 100 - 4 
121 - 25 
128 - 32 
196 - 100 
625 - 529 
97152 - 27
[74][492] - 482
772 - 183		 225 - 128 
2401 - 2304 
5929 - 5832 
9853 - 33
212 - 73		 125 - 27 
441 - 343 
9935 - 122
182 - 152
502 - [74][492]		 243 - 144 
324 - 225 
2500 - 2401 
10053 - 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


A000196 Prime Curios! Prime Puzzle
Wikipedia 196 Le nombre 196

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E-mail address : pdg@worldofnumbers.com