View source for Talk:First power of 2 that has leading decimal digits of 12

==License check==
We seem to be following the [https://projecteuler.net/copyright license] from the original site, (although I couldn't find the problem number to add a link).<br>
- Note: I am not the uthor of thie task, just checking the license. --[[User:Paddy3118|Paddy3118]] ([[User talk:Paddy3118|talk]]) 22:51, 15 January 2020 (UTC)

: I've found the [https://projecteuler.net/problem=686 problem number] which is 686. --[[User:PureFox|PureFox]] ([[User talk:PureFox|talk]]) 23:21, 15 January 2020 (UTC)

:: I was just about to add the (home page) information. &nbsp; I had originally phrased the problem much differently, but then it seemed to be more confusing and somewhat unclear, so I went with the original wording and phrasing that someone had paraphrased. &nbsp; I tried to find the problem on that website, but I couldn't find &nbsp; (I must have mis-remembered some keyword/keywords because I couldn't locate it via my search engine, other than it was on that site. &nbsp; &nbsp; -- [[User:Gerard Schildberger|Gerard Schildberger]] ([[User talk:Gerard Schildberger|talk]]) 23:29, 15 January 2020 (UTC)

:::Your RC title gives more info 👍 --[[User:Paddy3118|Paddy3118]] ([[User talk:Paddy3118|talk]]) 12:25, 16 January 2020 (UTC)

== significance of a given &nbsp; ''p'' &nbsp; value ==

::: I wonder what the significance is of saying that: "You are also given that p(123,45)=12710."? It's not as if it's going to take a long time to calculate whatever method is used. --[[User:PureFox|PureFox]] ([[User talk:PureFox|talk]]) 23:49, 15 January 2020 (UTC)

:::: The reasoning behind that (I believe) was to give the programmer an answer so that it could be used for verification of their programming solution to check if their calculation was working correctly for a "modest" (or easy) expression/formula. &nbsp; &nbsp; -- [[User:Gerard Schildberger|Gerard Schildberger]] ([[User talk:Gerard Schildberger|talk]]) 00:07, 16 January 2020 (UTC)

::::: Yeah, makes sense I guess. I'd been looking for a deeper meaning but couldn't think of any. --[[User:PureFox|PureFox]] ([[User talk:PureFox|talk]]) 00:12, 16 January 2020 (UTC)

== is there an easy way to get those delta 2^i (delta = i-Last_i ) ? ==

searching for '123' there is a double 485
Delta <BR>
<pre>   90  289  196  289  196  485  196  289  196  289  196  289  196  485  196  289  196  289  196  289  196  485  196  289  196  289  196  289  196  485  485  196 </pre>
searching for '317' ( not the smallest at start, 1651 seldom )
<pre>  779  485  485 1651  485  485 1166  485  485 1166  485  485 1166  485  485 1166  485  485 1166  485  485 1166  485  485 1651 </pre>
--[[User Horst.h|Horst.h]]12:14, 20 January 2020 (UTC)~

:In my first (abortive) attempt at this I'd noticed that (for a prefix of "123"), after an initial {90, 289} pair, subsequent differences were either 196, 289 or 485 (= 289 + 196). Moreover, 289 was always followed by 196 but 196 could be followed by either 289 or 485, and 485 could be followed by either 196 or 485 again. I wasn't able to discern any deeper pattern than this. So I'd ended up with this (Go) code:

:<lang go>func p123(n int) uint {
    power, shift := uint(0), uint(90)
    one := big.NewInt(1)
    temp := new(big.Int)
    for count := 0; count < n; count++ {
        power += shift
        switch shift {
        case 90:
            shift = 289
        case 289:
            shift = 196
        case 196:
            shift = 289
            temp.Set(one)
            temp.Lsh(temp, power+289)
            if !strings.HasPrefix(temp.String(), "123") {
                shift = 485
            }
        case 485:
            shift = 196
            temp.Set(one)
            temp.Lsh(temp, power+196)
            if !strings.HasPrefix(temp.String(), "123") {
                shift = 485
            }
        }
    }
    return power
}</lang>

:But, it turned out that this was very slow (4.5 minutes) as opposed to your log based approach at around 40 seconds or so. In his Perl 6 solution, I notice that Thundergnat has managed to combine your log approach with searching for patterns in the differences:

:<lang perl 6>my @startswith123 = lazy gather loop {
    state $pre   = '1.23';
    state $count = 0;
    state $this  = 0;
    given $this {
        when 196 {
            $this = 289;
            my \n = $count + $this;
            $this = 485 unless ( 10 ** (n * $ln2ln10 % 1) ).substr(0,4) eq $pre;
        }
        when 485 {
            $this = 196;
            my \n = $count + $this;
            $this = 485 unless ( 10 ** (n * $ln2ln10 % 1) ).substr(0,4) eq $pre;
        }
        when 289 { $this = 196 }
        when 90  { $this = 289 }
        when 0   { $this = 90  }
    }
    take $count += $this;
}</lang>

:though, given the substantial speed-up you've already achieved with your 'alternative' version, I don't know whether considering patterns as well is going to give a worthwhile improvement. --[[User:PureFox|PureFox]] ([[User talk:PureFox|talk]]) 14:47, 20 January 2020 (UTC)
::I counted the count of occurrences of deltas.There maximal 3 and one with 1 for the first occurrence.There many long existing deltas like 485/2136 and there are not so much different deltas. 
<pre>
The  9,999th  occurrence of 2 raised to a power whose product starts with "1" is 33,216
    3 6780|    4 3219|  // delta=3 6780 times and delta = 4 3219x 
The  9,999th  occurrence of 2 raised to a power whose product starts with "2" is 56,770
    1    1|    3 4496|    7 3913|   10 1589|
The  9,999th  occurrence of 2 raised to a power whose product starts with "3" is 80,047
    3 2241|    5    1|    7 1417|   10 6340|
The  9,999th  occurrence of 2 raised to a power whose product starts with "4" is 103,168
    2    1|   10 8936|   13 1062|
The  9,999th  occurrence of 2 raised to a power whose product starts with "5" is 126,269
    9    1|   10 8698|   23  362|   33  938|
The  9,999th  occurrence of 2 raised to a power whose product starts with "6" is 149,330
    6    1|   10 8460|   33  141|   43 1397|
The  9,999th  occurrence of 2 raised to a power whose product starts with "7" is 172,484
   10 8222|   43  391|   46    1|   53 1385|
The  9,999th  occurrence of 2 raised to a power whose product starts with "8" is 195,442
    3    1|   10 7985|   53 1123|   63  890|
The  9,999th  occurrence of 2 raised to a power whose product starts with "9" is 218,543
   10 7748|   53   75|   63 2176|
The  9,999th  occurrence of 2 raised to a power whose product starts with "10" is 241,491
   10 7512|   63 1518|   73  969|
The  9,999th  occurrence of 2 raised to a power whose product starts with "11" is 264,625
   10 7273|   50    1|   63  708|   73 2017|
The  9,999th  occurrence of 2 raised to a power whose product starts with "12" is 287,583
    7    1|   10 7036|   73 2863|   83   99|
The  9,999th  occurrence of 2 raised to a power whose product starts with "13" is 310,614
   10 6799|   17    1|   73 2291|   83  908|
The  9,999th  occurrence of 2 raised to a power whose product starts with "14" is 333,738
   10 6561|   47    1|   73 1719|   83 1718|
The  9,999th  occurrence of 2 raised to a power whose product starts with "15" is 356,799
   10 6324|   73 1146|   77    1|   83 2528|
The  9,999th  occurrence of 2 raised to a power whose product starts with "16" is 379,717
    4    1|   10 6087|   73  577|   83 3334|
The  9,999th  occurrence of 2 raised to a power whose product starts with "17" is 402,758
   10 5850|   34    1|   73    6|   83 4142|
The  9,999th  occurrence of 2 raised to a power whose product starts with "18" is 425,882
   10 5612|   54    1|   83 3819|   93  567|
The  9,999th  occurrence of 2 raised to a power whose product starts with "19" is 448,923
   10 5375|   83 3485|   84    1|   93 1138|
The  9,999th  occurrence of 2 raised to a power whose product starts with "20" is 471,758
    1    1|   10 5139|   83 3152|   93 1707|

... 83 vanishes 103 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "29" is 679,193
   10 3003|   68    1|   83  144|   93 6851|
The 9,999th occurrence of 2 raised to a power whose product starts with "30" is 702,214
   10 2766|   78    1|   93 7042|  103  190|
... 10 vanishes 196 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "41" is 955,269
   10  159|   22    1|   93 5976|  103 3863|
The 9,999th occurrence of 2 raised to a power whose product starts with "42" is 978,383
   32    1|   93 5879|  103 4040|  196   79|
... 103 vanishes 289 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "70" is 1,622,964
   46    1|   93 3165|  103  115|  196 6718|
The 9,999th occurrence of 2 raised to a power whose product starts with "71" is 1,646,274
   93 3067|  149    1|  196 6905|  289   26|
... 93 vanishes 485 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "102" is 2,360,602
   10    1|   93   59|  196 5562|  289 4377|
The 9,999th occurrence of 2 raised to a power whose product starts with "103" is 2,381,673
  113    1|  196 5522|  289 4447|  485   29|
... 289 vanishes 681 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "185" is 4,269,695
  196 1975|  260    1|  289   45|  485 7978|
The 9,999th occurrence of 2 raised to a power whose product starts with "186" is 4,293,872
  196 1930|  456    1|  485 8057|  681   11|
The 9,999th occurrence of 2 raised to a power whose product starts with "187" is 4,316,594
... 196 vanishes 1166 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "230" is 5,306,685
   61    1|  196   26|  485 7599|  681 2373|
The 9,999th occurrence of 2 raised to a power whose product starts with "231" is 5,329,696
  257    1|  485 7589|  681 2392| 1166   17|
... 681 vanishes 1651 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "303" is 6,987,488
  274    1|  485 6839|  681   27| 1166 3132|
The 9,999th occurrence of 2 raised to a power whose product starts with "304" is 7,010,014
  470    1|  485 6829| 1166 3164| 1651    5|
... 1166 vanishes 2136 arises
  The 9,999th occurrence of 2 raised to a power whose product starts with "444" is 10,236,012
  485 5371| 1166   20| 1600    1| 1651 4607|
The 9,999th occurrence of 2 raised to a power whose product starts with "445" is 10,257,372
  485 5361|  630    1| 1651 4635| 2136    2|
... 1651 vanishes 2621 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "830" is 19,117,769
  116    1|  485 1355| 1651    2| 2136 8641|
The 9,999th occurrence of 2 raised to a power whose product starts with "831" is 19,143,401
  485 1344| 1767    1| 2136 8644| 2621   10|
... 485 vanishes 4757 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "959" is 22,091,629
  485   11| 1342    1| 2136 8435| 2621 1552|
The 9,999th occurrence of 2 raised to a power whose product starts with "960" is 22,115,610
  857    1| 2136 8433| 2621 1565|
The 9,999th occurrence of 2 raised to a power whose product starts with "961" is 22,130,562
  372    1| 2136 8432| 2621 1559| 4757    7|
 ##### 4 Digits
... 2621 vanishes 6893 arises
  The 9,999th occurrence of 2 raised to a power whose product starts with "1,138" is 26,210,262
 1608    1| 2136 8144| 2621    3| 4757 1851|
The 9,999th occurrence of 2 raised to a power whose product starts with "1,139" is 26,234,243
 1123    1| 2136 8142| 4757 1850| 6893    6|
... 4757 vanishes 9029 arises
The 9,999th occurrence of 2 raised to a power whose product starts with "1,397" is 32,170,665
 2136 7723| 3442    1| 4757    5| 6893 2270|
The 9,999th occurrence of 2 raised to a power whose product starts with "1,398" is 32,192,510
  821    1| 2136 7721| 6893 2275| 9029    2|
... 6893 vanishes 11165 arises 
 The 9,999th occurrence of 2 raised to a power whose product starts with "1,809" is 41,649,940
  250    1| 2136 7052| 6893    6| 9029 2940|
The 9,999th occurrence of 2 raised to a power whose product starts with "1,810" is 41,685,086
 2136 7050| 6658    1| 9029 2947|11165    1|
... 9029 vanishes 13301 arises  
The 9,999th occurrence of 2 raised to a power whose product starts with "2,566" is 59,093,850
 2136 5818| 3310    1| 9029    3|11165 4177|
The 9,999th occurrence of 2 raised to a power whose product starts with "2,567" is 59,113,559
 2136 5817| 7582    1|11165 4181|
The 9,999th occurrence of 2 raised to a power whose product starts with "2,568" is 59,122,103
  689    1| 2136 5816|11165 4179|13301    3|
... 11165 vanishes 15437 arises    
The 9,999th occurrence of 2 raised to a power whose product starts with "4,412" is 101,586,005
  723    1| 2136 2812|11165    1|13301 7185|
The 9,999th occurrence of 2 raised to a power whose product starts with "4,413" is 101,603,578
 2136 2811| 2859    1|13301 7186|15437    1|  
... 2136 vanishes 12354 arises  
The 9,999th occurrence of 2 raised to a power whose product starts with "6,138" is 141,314,661
 2136    2| 3969    1|13301 6087|15437 3909|
The 9,999th occurrence of 2 raised to a power whose product starts with "6,139" is 141,343,399
 6105    1|13301 6087|15437 3911|
The 9,999th occurrence of 2 raised to a power whose product starts with "6,140" is 141,374,273
 8241    1|12354    2|13301 6086|15437 3910|
....
The 9,999th occurrence of 2 raised to a power whose product starts with "9,999" is 230,241,843
12354 6288|13301 3628|15437   83|</pre>
--[[User:Horst.h]]19:22, 20 January 2020 (UTC)~

:Taking the sequence Log<sub>10</sub> 2..Log<sub>10</sub> 2..infinity, then the task is to find values of this sequence whose decimal part is greater than or equal to the decimal part of Log<sub>10</sub> 1.2 and less than the decimal part of Log<sub>10</sub> 1.3. This can of course be calculated without considering every value of Log<sub>10</sub> 2..Log<sub>10</sub> 2..infinity.--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 15:11, 15 March 2021 (UTC)

==Why does the title limit the task to starting 12?==

Many of the solutions solve the general case of starting say 99 etc. There is a particular significance to 12 but the task description does not emphasize this.--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 14:59, 15 March 2021 (UTC)

: What would you suggest to emphasize that particular significance? &nbsp; &nbsp; -- [[User:Gerard Schildberger|Gerard Schildberger]] ([[User talk:Gerard Schildberger|talk]]) 17:18, 15 March 2021 (UTC)