Powertrains

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Relue
Posts: 6
Joined: Wed Jun 27, 2012 3:00 pm UTC
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Powertrains

Postby Relue » Mon Aug 10, 2015 12:37 pm UTC

In Alex's Adventures in Numberland, John Horton Conway's concept of
powertrains is mentioned on p.261.

For any number written abcdefg..., its 'powertrain' is (a^b).(c^d).(e^f).g...,
with 0^0 = 1. So 2015 -> (2^0).(1^5) = 1.1 = 1.

Repeatedly reapplying a powertrain reduces almost every number to a single digit.
Those which don't are called 'indestructible digits'.

Two numbers are cited as being indestructible, 2592, found by Conway, and
24547284284866560000000000, found by Neil Sloane. These are the only two
known to exist.

2592 -> (2^5).(9^2) = 32.81 = 2592.

So far so good. But what about 2534?

2534 -> (2^5).(3^4) = 32.81 = 2592 -> 2592.

Doesn't that also make 2534 indestructible according to the definition,
or have I misunderstood something?

Does the word 'reduce' mean that only powertrains with successively smaller
outputs are permissible? From the text, the implication is that this isn't
the case. Numbers ending in 99 with an even number of digits all output
much higher numbers before collapsing to a single digit.

>-)
Posts: 525
Joined: Tue Apr 24, 2012 1:10 am UTC

Re: Powertrains

Postby >-) » Mon Aug 10, 2015 3:37 pm UTC

yeah if you take that definition of indestructible then 2534 is, but the book specifically mentions the two numbers as fixed points, which 2534 would not be.

also reduce doesn't necessarily mean to reduce in size in general, mathematically speaking. you can say P(2534) reduces to 2592 in the same way sin/cos reduces to tan.

TurboSpencer96
Posts: 1
Joined: Fri Mar 16, 2018 8:28 pm UTC

Re: Powertrains

Postby TurboSpencer96 » Fri Mar 16, 2018 8:44 pm UTC

Dear Relue,

I have read same book and I was also wondering at same problem. It was some years ago, but now, when I am older I used my computer and made program to calculate this phenomenon. John Horton Conway and his friend mentioned only about these two numbers 2592 and 24547284284866560000000000 because they are undestroyable at first time of powertrain use (sorry for my English, it's not my native language). As I said I made a program what can find all no-powertrain numbers as far as long it continue calculating. There are some of them:
Spoiler:
642
2164
2534
2592
3425
6421
9225
10642
11642
12642
13642
14642
15642
16642
17642
18642
19642
20642
21636
21641
22348
22633
22928
23344
23629
23924
24336
24342
24922
25329
25333
25341
25919
25921
30642
31634
32259
32628
33246
33253
33426
34228
34234
34242
34251
34418
34422
34814
40642
41348
41633
41928
42336
42342
42922
50642
52648
60642
61632
62239
62328
62622
62819
62918
63216
63223
63314
63413
63612
63724
64002
64102
64112
64122
64132
64142
64152
64162
64172
64182
64192
64202
64211
64302
64402
64502
64528
64602
64702
64802
64902
67792
69996
70642
72634
79672
80642
81344
81629
81924
90642
91259
91628
92228
92234
92242
92251
92418
92422
92814
99696
102164
102534
102592
103425
106421
109225
112164
112534
112592
113425
116421
119225
122164
122534
122592
123425
126421
129225
132164
132534
132592
133425
136421
139225
142164
142534
142592
143425
146421
149225
152164
152534
152592
153425
156421
159225
162164
162534
162592
163425
166421
169225
172164
172534
172592
173425
176421
179225
182164
182534
182592
183425
186421
189225
192164
192534
192592
193425
196421
199225
202164
202534
202592
203425
206421
209225
210064
211064
211164
211264
211364
211464
211564
211664
211764
211864
211964
212064
212434
212492
213064
213424
213442
214064
214234
214292
215064
216064
216163
216262
216361
216400
216410
216411
216412
216413
216414
216415
216416
216417
216418
216419
216420
216430
216440
216450
216460
216470
216480
216490
216779
217064
217967
218064
219064
219224
219242
222334
222392
223163
223423
223481
226331
226372
227263
228134
228192
229223
229281
232234
232292
233262
233422
233441
234134
234192
235264
236232
236291
236452
239162
239222
239241
242134
242192
243361
243421
246133
249221
250034
250092
251034
251092
251134
251192
251234
251292
251334
251392
251434
251492
251534
251592
251634
251692
251734
251792
251834
251892
251934
251992
252034
252092
253034
253092
253133
253232
253291
253331
253372
253400
253410
253411
253412
253413
253414
253415
253416
253417
253418
253419
253420
253430
253440
253450
253460
253470
253480
253490
254034
254092
255034
255092
256034
256092
257034
257092
257233
258034
258092
259034
259092
259132
259191
259200
259210
259211
259212
259213
259214
259215
259216
259217
259218
259219
259220
259230
259240
259250
259260
259270
259280
259290
273452
275234
275292
279252
283779
287937
302164
302534
302592
303425
306421
309225
312263
312533
313325
314163
316322
316341
322362
322532
322591
323225
326223
326281
328162
329125
332461
332531
332572
333125
334261
336124
336142
337225
340025
341025
341125
341225
341325
341425
341525
341625
341725
341825
341925
342025
342124
342142
342223
342281
342322
342341
342421
342500
342510
342511
342512
342513
342514
342515
342516
342517
342518
342519
342520
342530
342540
342550
342560
342570
342580
342590
342752
343025
344025
344123
344181
344221
345025
345227
346025
347025
348025
348122
348141
349025
372879
374479
377928
377944
402164
402534
402592
403425
406421
409225
412334
412392
413163
413423
413481
416331
416372
417263
418134
418192
419223
419281
422134
422192
423361
423421
426133
429221
443779
447937
459599
459698
459797
459896
459995
502164
502534
502592
503425
506421
509225
522364
522734
522792
523427
526423
526481
528164
529227
596989
598969
602164
602534
602592
603425
606421
609225
612163
612433
613324
613342
614233
616321
616999
619969
622162
622332
622391
623223
623281
626221
626899
628132
628191
629123
629181
629968
632161
632231
632272
633122
633141
634131
634172
636121
636799
637222
637241
639967
640021
641021
641121
641221
641321
641421
641521
641621
641721
641821
641921
642021
642100
642110
642111
642112
642113
642114
642115
642116
642117
642118
642119
642120
642130
642140
642150
642160
642170
642180
642190
642352
643021
644021
645021
645223
645281
646021
646699
647021
647888
648021
648152
648878
649021
649966
656599
659965
666499
669964
672179
676399
677921
679963
686299
689962
695989
696199
698959
699961
702164
702534
702592
703425
706421
709225
722263
722533
723325
724163
726322
726341
786488
788864
792167
792837
793728
793744
794437
796721
802164
802534
802592
...

I think there's infinity numbers what are invulnerable for powertrain with more than one use. All these numbers goes to 2592, and maybe, if I'll find higher numbers than 24547284284866560000000000 probably they also reduces to that number, not only 2592. But these two numbers absolutely stands out.

Best regards for all who have read my post


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