World!Of
Numbers

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196 |

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

( 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).
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

A000196 Prime Curios! Prime Puzzle
Wikipedia 196 Le nombre 196
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