Re: Mr.Bean's Conjecture

[a1call]

FWIW my ¢2:

The fact that it has not been proven to be without a solution so far is a strong indication that the probability of the existence of a solution is greater than 0.

Yet due to the nurture of the problem, the probability of finding a solution decreases exponentially as the parameters get larger.

Normally with a constant probability, no matter how low, infinite trials is guaranteed to produce a solution.

This got me thinking. Could the probability of a solution decreases so rapidly (for larger parameters) that it would make it impossible for a solution to exist?

I would say not.

World appreciate your thoughts.
Thank you in advance.

[retina]

Originally Posted by a1call

... the probability of a solution ...

A solution either exists, or it does not. How is probability involved here?

[a1call]

Originally Posted by retina

A solution either exists, or it does not. How is probability involved here?

Well, that's a million dollar question. Literally.

https://en.m.wikipedia.org/wiki/Beal_conjecture

[a1call]

Perhaps the following similar concepts would clarify:

print("\nBVC-100-A-Forum.gp\nBy a1call")

initialProbability = 2
k=19
i=0
while(k!=1,{
  i=i+1;
  k=random(initialProbability *i);
  print("** ",i," -> ",initialProbability *i," -> ", k);
  \\if (i>19,next(19));
})

print("**** Found a solution in ",i, " trials.")

The following results are from consecutively increasing initialProbability:

BVC-100-A-Forum.gp
By a1call
2
19
0
** 1 -> 2 -> 1
**** Found a solution in 1 trials.

BVC-100-A-Forum.gp
By a1call
3
19
0
** 1 -> 3 -> 2
** 2 -> 6 -> 3
** 3 -> 9 -> 5
** 4 -> 12 -> 9
** 5 -> 15 -> 12
** 6 -> 18 -> 17
** 7 -> 21 -> 17
** 8 -> 24 -> 18
** 9 -> 27 -> 11
** 10 -> 30 -> 3
** 11 -> 33 -> 3
** 12 -> 36 -> 14
** 13 -> 39 -> 29
** 14 -> 42 -> 18
** 15 -> 45 -> 14
** 16 -> 48 -> 1
**** Found a solution in 16 trials.

BVC-100-A-Forum.gp
By a1call
4
19
0
** 1 -> 4 -> 2
** 2 -> 8 -> 3
** 3 -> 12 -> 5
** 4 -> 16 -> 9
** 5 -> 20 -> 17
** 6 -> 24 -> 17
** 7 -> 28 -> 18
** 8 -> 32 -> 11
** 9 -> 36 -> 7
** 10 -> 40 -> 3
** 11 -> 44 -> 14
** 12 -> 48 -> 29
** 13 -> 52 -> 18
** 14 -> 56 -> 14
** 15 -> 60 -> 1
**** Found a solution in 15 trials.

BVC-100-A-Forum.gp
By a1call
5
19
0
** 1 -> 5 -> 3
** 2 -> 10 -> 5
** 3 -> 15 -> 9
** 4 -> 20 -> 17
** 5 -> 25 -> 17
** 6 -> 30 -> 18
** 7 -> 35 -> 23
** 8 -> 40 -> 7
** 9 -> 45 -> 3
** 10 -> 50 -> 14
** 11 -> 55 -> 29
** 12 -> 60 -> 18
** 13 -> 65 -> 28
** 14 -> 70 -> 3
** 15 -> 75 -> 19
** 16 -> 80 -> 49
** 17 -> 85 -> 57
** 18 -> 90 -> 68
** 19 -> 95 -> 43
** 20 -> 100 -> 95
** 21 -> 105 -> 6
** 22 -> 110 -> 74
** 23 -> 115 -> 57
** 24 -> 120 -> 68
** 25 -> 125 -> 66
** 26 -> 130 -> 4
** 27 -> 135 -> 8
** 28 -> 140 -> 90
** 29 -> 145 -> 2
** 30 -> 150 -> 146
** 31 -> 155 -> 37
** 32 -> 160 -> 146
** 33 -> 165 -> 29
** 34 -> 170 -> 68
** 35 -> 175 -> 3
** 36 -> 180 -> 150
** 37 -> 185 -> 86
** 38 -> 190 -> 137
** 39 -> 195 -> 180
** 40 -> 200 -> 88
** 41 -> 205 -> 4
** 42 -> 210 -> 209
** 43 -> 215 -> 67
** 44 -> 220 -> 215
** 45 -> 225 -> 143
** 46 -> 230 -> 215
** 47 -> 235 -> 43
** 48 -> 240 -> 95
** 49 -> 245 -> 163
** 50 -> 250 -> 25
** 51 -> 255 -> 5
** 52 -> 260 -> 118
** 53 -> 265 -> 98
** 54 -> 270 -> 134
** 55 -> 275 -> 273
** 56 -> 280 -> 175
** 57 -> 285 -> 13
** 58 -> 290 -> 123
** 59 -> 295 -> 250
** 60 -> 300 -> 250
** 61 -> 305 -> 66
** 62 -> 310 -> 143
** 63 -> 315 -> 274
** 64 -> 320 -> 155
** 65 -> 325 -> 299
** 66 -> 330 -> 180
** 67 -> 335 -> 229
** 68 -> 340 -> 130
** 69 -> 345 -> 248
** 70 -> 350 -> 271
** 71 -> 355 -> 24
** 72 -> 360 -> 214
** 73 -> 365 -> 341
** 74 -> 370 -> 128
** 75 -> 375 -> 139
** 76 -> 380 -> 247
** 77 -> 385 -> 167
** 78 -> 390 -> 213
** 79 -> 395 -> 377
** 80 -> 400 -> 320
** 81 -> 405 -> 203
** 82 -> 410 -> 373
** 83 -> 415 -> 152
** 84 -> 420 -> 278
** 85 -> 425 -> 68
** 86 -> 430 -> 351
** 87 -> 435 -> 412
** 88 -> 440 -> 378
** 89 -> 445 -> 407
** 90 -> 450 -> 55
** 91 -> 455 -> 65
** 92 -> 460 -> 64
** 93 -> 465 -> 420
** 94 -> 470 -> 124
** 95 -> 475 -> 117
** 96 -> 480 -> 141
** 97 -> 485 -> 57
** 98 -> 490 -> 415
** 99 -> 495 -> 79
** 100 -> 500 -> 66
** 101 -> 505 -> 243
** 102 -> 510 -> 154
** 103 -> 515 -> 351
** 104 -> 520 -> 268
** 105 -> 525 -> 21
** 106 -> 530 -> 161
** 107 -> 535 -> 135
** 108 -> 540 -> 452
** 109 -> 545 -> 467
** 110 -> 550 -> 39
** 111 -> 555 -> 479
** 112 -> 560 -> 456
** 113 -> 565 -> 380
** 114 -> 570 -> 186
** 115 -> 575 -> 207
** 116 -> 580 -> 239
** 117 -> 585 -> 309
** 118 -> 590 -> 345
** 119 -> 595 -> 481
** 120 -> 600 -> 411
** 121 -> 605 -> 420
** 122 -> 610 -> 439
** 123 -> 615 -> 358
** 124 -> 620 -> 74
** 125 -> 625 -> 512
** 126 -> 630 -> 438
** 127 -> 635 -> 33
** 128 -> 640 -> 163
** 129 -> 645 -> 278
** 130 -> 650 -> 355
** 131 -> 655 -> 179
** 132 -> 660 -> 308
** 133 -> 665 -> 352
** 134 -> 670 -> 440
** 135 -> 675 -> 421
** 136 -> 680 -> 31
** 137 -> 685 -> 21
** 138 -> 690 -> 592
** 139 -> 695 -> 256
** 140 -> 700 -> 195
** 141 -> 705 -> 166
** 142 -> 710 -> 363
** 143 -> 715 -> 562
** 144 -> 720 -> 504
** 145 -> 725 -> 120
** 146 -> 730 -> 463
** 147 -> 735 -> 353
** 148 -> 740 -> 660
** 149 -> 745 -> 673
** 150 -> 750 -> 94
** 151 -> 755 -> 39
** 152 -> 760 -> 592
** 153 -> 765 -> 519
** 154 -> 770 -> 291
** 155 -> 775 -> 229
** 156 -> 780 -> 765
** 157 -> 785 -> 281
** 158 -> 790 -> 663
** 159 -> 795 -> 106
** 160 -> 800 -> 490
** 161 -> 805 -> 548
** 162 -> 810 -> 623
** 163 -> 815 -> 642
** 164 -> 820 -> 343
** 165 -> 825 -> 231
** 166 -> 830 -> 520
** 167 -> 835 -> 196
** 168 -> 840 -> 644
** 169 -> 845 -> 383
** 170 -> 850 -> 169
** 171 -> 855 -> 486
** 172 -> 860 -> 236
** 173 -> 865 -> 724
** 174 -> 870 -> 87
** 175 -> 875 -> 685
** 176 -> 880 -> 783
** 177 -> 885 -> 675
** 178 -> 890 -> 814
** 179 -> 895 -> 259
** 180 -> 900 -> 132
** 181 -> 905 -> 696
** 182 -> 910 -> 177
** 183 -> 915 -> 393
** 184 -> 920 -> 472
** 185 -> 925 -> 679
** 186 -> 930 -> 464
** 187 -> 935 -> 242
** 188 -> 940 -> 192
** 189 -> 945 -> 788
** 190 -> 950 -> 580
** 191 -> 955 -> 825
** 192 -> 960 -> 344
** 193 -> 965 -> 672
** 194 -> 970 -> 210
** 195 -> 975 -> 516
** 196 -> 980 -> 297
** 197 -> 985 -> 318
** 198 -> 990 -> 944
** 199 -> 995 -> 345
** 200 -> 1000 -> 583
** 201 -> 1005 -> 689
** 202 -> 1010 -> 819
** 203 -> 1015 -> 757
** 204 -> 1020 -> 718
** 205 -> 1025 -> 942
** 206 -> 1030 -> 857
** 207 -> 1035 -> 677
** 208 -> 1040 -> 102
** 209 -> 1045 -> 828
** 210 -> 1050 -> 733
** 211 -> 1055 -> 75
** 212 -> 1060 -> 597
** 213 -> 1065 -> 36
** 214 -> 1070 -> 500
** 215 -> 1075 -> 222
** 216 -> 1080 -> 273
** 217 -> 1085 -> 626
** 218 -> 1090 -> 826
** 219 -> 1095 -> 795
** 220 -> 1100 -> 705
** 221 -> 1105 -> 352
** 222 -> 1110 -> 550
** 223 -> 1115 -> 41
** 224 -> 1120 -> 580
** 225 -> 1125 -> 742
** 226 -> 1130 -> 871
** 227 -> 1135 -> 746
** 228 -> 1140 -> 760
** 229 -> 1145 -> 429
** 230 -> 1150 -> 531
** 231 -> 1155 -> 20
** 232 -> 1160 -> 336
** 233 -> 1165 -> 643
** 234 -> 1170 -> 102
** 235 -> 1175 -> 998
** 236 -> 1180 -> 704
** 237 -> 1185 -> 582
** 238 -> 1190 -> 982
** 239 -> 1195 -> 83
** 240 -> 1200 -> 114
** 241 -> 1205 -> 616
** 242 -> 1210 -> 580
** 243 -> 1215 -> 798
** 244 -> 1220 -> 1104
** 245 -> 1225 -> 1070
** 246 -> 1230 -> 726
** 247 -> 1235 -> 568
** 248 -> 1240 -> 900
** 249 -> 1245 -> 97
** 250 -> 1250 -> 645
** 251 -> 1255 -> 518
** 252 -> 1260 -> 278
** 253 -> 1265 -> 1203
** 254 -> 1270 -> 703
** 255 -> 1275 -> 1122
** 256 -> 1280 -> 1
**** Found a solution in 256 trials.

** 252 -> 1512 -> 278
** 253 -> 1518 -> 1203
** 254 -> 1524 -> 703
** 255 -> 1530 -> 1122
** 256 -> 1536 -> 1
**** Found a solution in 256 trials.

** 254 -> 1778 -> 703
** 255 -> 1785 -> 1122
** 256 -> 1792 -> 1632
** 257 -> 1799 -> 1640
** 258 -> 1806 -> 1597
** 259 -> 1813 -> 1
**** Found a solution in 259 trials.

** 99331 -> 794648 -> 474659
** 99332 -> 794656 -> 618593
** 99333 -> 794664 -> 1
**** Found a solution in 99333 trials.

** 101759 -> 915831 -> 474659
** 101760 -> 915840 -> 618593
** 101761 -> 915849 -> 1
**** Found a solution in 101761 trials.

.....

initialProbability = 12

** 165391 -> 1984692 -> 113355
** 165392 -> 1984704 -> 821599
** 165393 -> 1984716 -> 1217526
** 165394 -> 1984728 -> 1760329
** 165395 -> 1984740 -> 1231605
** 165396 -> 1984752 -> 1
**** Found a solution in 165396 trials.

initialProbability = 13

** 4687726 -> 60940438 -> 29800996
** 4687727 -> 60940451 -> 56143258
** 4687728 -> 60940464 -> 46069460
** 4687729 -> 60940477 -> 13719312
^C** 4687730 -> 60940490 -> 53141080
  ***   at top-level: ...om(initialProbability*i);print("** ",i," -> "
  ***                                             ^--------------------
  *** print: user interrupt after 14,836 ms
**** Found a solution in i trials.

[VBCurtis]

Originally Posted by a1call

Well, that's a million dollar question. Literally.

https://en.m.wikipedia.org/wiki/Beal_conjecture

You didn't answer his question, so I'll repeat it: How is probability involved here?

Are you saying you can compute a probability of a solution? If so, doing a calculation that shows the probability goes to zero as the tested-limits increase is *not* a proof that a solution does not exist. Your what-if ponderings indicate a fundamental lack of grasp of what probability means; a cynic might say that you think not finding a solution means we won't find a solution, and further that you think we can "prove" we won't find a solution based on previous failed attempts. That's not probability at all, nor is it a proof; it's verbal hand-waving of no substance whatsoever.

[a1call]

Another clearer way of expressing the question in OP is:

Is there a finite value for initialProbability in the above code, for which no solution will ever be found, even if you run the code indefinitely/for-infinitum.

[retina]

Originally Posted by a1call

Another clearer way of expressing the question in OP is:

Is there a finite value for initialProbability in the above code, for which no solution will ever be found, even if you run the code indefinitely/for-infinitum.

Yes, of course: initialProbability <= 0

But what does that have to do with Beal's Conjecture?

BTW: initialProbability is very badly named. Probabilities can't be more than 1 or less than 0.

[a1call]

Originally Posted by retina

Yes, of course: initialProbability <= 0

But what does that have to do with Beal's Conjecture?

BTW: initialProbability is very badly named. Probabilities can't be more than 1 or less than 0.

the random function is valid only for numbers greater than 0. So It goes without saying that the initialProbability will have to be an integer greater tan 0.

You are correct however about a poor choice of words for the variable.

A more accurate name would be inverseOfTheInitiakProbabilityMinusOne but I was not going to give my variable such an unnecessarily long name.

The relation to the concept to finding a solution to Beal's conjecture should be obvious to most. I don't think it is necessary to elaborate any further.

If you don't agree, please feel free to let people who can see the relationship respond instead.
Thank you for all the replies.

[a1call]

BTW, I believe the answer to my post number 6 is No, as pointed out in OP.

[retina]

Originally Posted by a1call

BTW, I believe the answer to my post number 6 is No, as pointed out in OP.

That depends upon the quality of your random generator. What implementation are you using?

[retina]

Originally Posted by a1call

The relation to the concept to finding a solution to Beal's conjecture should be obvious to most. I don't think it is necessary to elaborate any further.

If you don't agree, please feel free to let people who can see the relationship respond instead.

I wanted you to explain the relevance of your code. I find sometimes people can see insight when asked to explain things to others.