(A) For a base ten positive integer P drawn at random between 10 and 99 inclusively, determine the probability that the first two digits (reading left to right) in the base ten expansion of 2^P is equal to P-1.

(B) For a base ten positive integer P drawn at random between 10 and 99 inclusively, determine the probability that the first two digits (reading left to right) in the base ten expansion of 6^P is equal to P-1.

(In reply to re(2): exploration turned up something strange by Jer)

Well, for such an exension, here's a start:

The integral powers of 2 are listed, and an asterisk is placed next to the beginning of an interval in which a solution is possible, regarding the integer part of P. For example 2^13 = 8192 is the first such marked range, as between 8192 and 16,192 there lies a number that is 2^13+ (the + representing a fraction < 1) where the first two digits are 12. In this case, the range of numbers that begin with 12 is [12,000 to 13,000) where the square bracket denotes that the first number is inclusive (a closed end to the set) and the second is exclusive (an open end to the set). Of course the power of 2, P, is found by taking the base 2 logarithm of these numbers. The range turns out to be approximately 13.5507467853832428865 to 13.666224002803178252681, the base 2 logs of 12,000 and 13,000 respectively.  Similar subranges could be found for each of the starred ranges, such as between 131,072 and 262,144 there's a range 160,000 to 169,999.999... where the base 2 logs can be found for a range of values of P that are 17+ and lead to these values that start with 16.

10  1024
11  2048
12  4096
13  8192 *
14  16384
15  32768
16  65536
17  131072 *
18  262144
19  524288
20  1048576 *
21  2097152 *
22  4194304 
23  8388608
24  16777216 *
25  33554432
26  67108864
27  134217728 *
28  268435456 *
29  536870912 
30  1073741824
31  2147483648 *
32  4294967296
33  8589934592
34  17179869184 *
35  34359738368 *
36  68719476736
37  137438953472
38  274877906944 *
39  549755813888
40  1099511627776
41  2199023255552 *
42  4398046511104
43  8796093022208
44  17592186044416 
45  35184372088832 *
46  70368744177664
47  140737488355328
48  281474976710656 *
49  562949953421312
50  1125899906842624 
51  2251799813685248
52  4503599627370496 *
53  9007199254740992
54  18014398509481984
55  36028797018963968 *
56  72057594037927936
57  144115188075855872
58  288230376151711744 *
59  576460752303423488 *
60  1152921504606846976
61  2305843009213693952
62  4611686018427387904 *
63  9223372036854775808
64  18446744073709551616
65  36893488147419103232 *
66  73786976294838206464 
67  147573952589676412928
68  295147905179352825856
69  590295810358705651712 *
70  1180591620717411303424
71  2361183241434822606848
72  4722366482869645213696 *
73  9444732965739290427392
74  18889465931478580854784
75  37778931862957161709568 *
76  75557863725914323419136 *
77  151115727451828646838272 *
78  302231454903657293676544
79  604462909807314587353088 *
80  1208925819614629174706176
81  2417851639229258349412352
82  4835703278458516698824704 *
83  9671406556917033397649408
84  19342813113834066795298816
85  38685626227668133590597632
86  77371252455336267181195264 *
87  154742504910672534362390528 
88  309485009821345068724781056
89  618970019642690137449562112 *
90  1237940039285380274899124224
91  2475880078570760549798248448
92  4951760157141521099596496896 *
93  9903520314283042199192993792 
94  19807040628566084398385987584
95  39614081257132168796771975168
96  79228162514264337593543950336 *
97  158456325028528675187087900672
98  316912650057057350374175801344
99  633825300114114700748351602688

For the powers of 6, I'll leave it to you to mark the ranges in which to look for such subranges:

10  60466176
11  362797056
12  2176782336
13  13060694016
14  78364164096
15  470184984576
16  2821109907456
17  16926659444736
18  101559956668416
19  609359740010496
20  3656158440062976
21  21936950640377856
22  131621703842267136
23  789730223053602816
24  4738381338321616896
25  28430288029929701376
26  170581728179578208256
27  1023490369077469249536
28  6140942214464815497216
29  36845653286788892983296
30  221073919720733357899776
31  1326443518324400147398656
32  7958661109946400884391936
33  47751966659678405306351616
34  286511799958070431838109696
35  1719070799748422591028658176
36  10314424798490535546171949056
37  61886548790943213277031694336
38  371319292745659279662190166016
39  2227915756473955677973140996096
40  13367494538843734067838845976576
41  80204967233062404407033075859456
42  481229803398374426442198455156736
43  2887378820390246558653190730940416
44  17324272922341479351919144385642496
45  103945637534048876111514866313854976
46  623673825204293256669089197883129856
47  3742042951225759540014535187298779136
48  22452257707354557240087211123792674816
49  134713546244127343440523266742756048896
50  808281277464764060643139600456536293376
51  4849687664788584363858837602739217760256
52  29098125988731506183153025616435306561536
53  174588755932389037098918153698611839369216
54  1047532535594334222593508922191671036215296
55  6285195213566005335561053533150026217291776
56  37711171281396032013366321198900157303750656
57  226267027688376192080197927193400943822503936
58  1357602166130257152481187563160405662935023616
59  8145612996781542914887125378962433977610141696
60  48873677980689257489322752273774603865660850176
61  293242067884135544935936513642647623193965101056
62  1759452407304813269615619081855885739163790606336
63  10556714443828879617693714491135314434982743638016
64  63340286662973277706162286946811886609896461828096
65  380041719977839666236973721680871319659378770968576
66  2280250319867037997421842330085227917956272625811456
67  13681501919202227984531053980511367507737635754868736
68  82089011515213367907186323883068205046425814529212416
69  492534069091280207443117943298409230278554887175274496
70  2955204414547681244658707659790455381671329323051646976
71  17731226487286087467952245958742732290027975938309881856
72  106387358923716524807713475752456393740167855629859291136
73  638324153542299148846280854514738362441007133779155746816
74  3829944921253794893077685127088430174646042802674934480896
75  22979669527522769358466110762530581047876256816049606885376
76  137878017165136616150796664575183486287257540896297641312256
77  827268102990819696904779987451100917723545245377785847873536
78  4963608617944918181428679924706605506341271472266715087241216
79  29781651707669509088572079548239633038047628833600290523447296
80  178689910246017054531432477289437798228285773001601743140683776
81  1072139461476102327188594863736626789369714638009610458844102656
82  6432836768856613963131569182419760736218287828057662753064615936
83  38597020613139683778789415094518564417309726968345976518387695616
84  231582123678838102672736490567111386503858361810075859110326173696
85  1389492742073028616036418943402668319023150170860455154661957042176
86  8336956452438171696218513660416009914138901025162730927971742253056
87  50021738714629030177311081962496059484833406150976385567830453518336
88  300130432287774181063866491774976356909000436905858313406982721110016
89  1800782593726645086383198950649858141454002621435149880441896326660096
90  10804695562359870518299193703899148848724015728610899282651377959960576
91  64828173374159223109795162223394893092344094371665395695908267759763456
92  388969040244955338658770973340369358554064566229992374175449606558580736
93  2333814241469732031952625840042216151324387397379954245052697639351484416
94  14002885448818392191715755040253296907946324384279725470316185836108906496
95  84017312692910353150294530241519781447677946305678352821897115016653438976
96  504103876157462118901767181449118688686067677834070116931382690099920633856
97  3024623256944772713410603088694712132116406067004420701588296140599523803136
98  18147739541668636280463618532168272792698436402026524209529776843597142818816
99  108886437250011817682781711193009636756190618412159145257178661061582856912896

 

Posted by Charlie on 2011-03-17 15:29:19